摘要:

Relaxation in Simple Liquids by Polarized Light Scattering DORFMULLER FYTAS MERSCHAND DIMITRIOS BYTHOMAS *GEORGE WERNER SAMIOS Fakultat fur Chemie Universitat Bielefeld Morgenbreede 45 4800 Bielefeld 1 W. Germany Received 3rd August 1976 The polarized spectra of liquid CClr CH13 C6H5F and CsH5Cl were obtained by interferometric light scattering analysis over a temperature range of -80°C for each substance. The data obtained from the BriIlouin lines show a hypersound dispersion which could be fitted to a relaxation equation with a single relaxation time. We thus could obtain values for the relaxation time z and for the relaxation strength R as functions of the temperature. The values of R and the temperature dependence of this quantity have yielded an insight into the nature of the observed relaxation processes which appear to correspond to the T-V energy transfer involving the lowest or occasionally the second lowest vibrational level vinit.The temperature dependence of z enables us within the frame of the Schwarz-Slawsky-Herzfeld-Tanczos theory to obtain information about the inelastic collision cross section the steepness of the intermolecular repulsion and the ratio ttv/qs. The outcome of these calculations depends upon a num-ber of assumptions which have been discussed at some length. A macroscopic system is able to dissipate mechanical energy if the state variables are changed at a rate comparable to one of the relaxation times describing the kinetics of energy transfer between any of the degrees of freedom of the system.At high transfer rates the excitation of the process is obtained either by using ultrasonic or hypersonic waves. In the context of this study we shall have to deal with T-V relaxation i.e. with the time dependence of the excitation of molecular vibrations from the thermal pool of the translations. On the molecular level the T-V relaxation is connected with the time-dependent perturbations of the molecular levels by inter- molecular interactions. Specifically when we deal with gases the equation of state enters the calculations via the average time between binary collisions zBCand the duration of the collision zc. Under such simple conditions we may define a relaxation probability P’per second. This can be expressed as a product of a transition probability per collision P and r-lBc P’= P/rBC or P’=Z/Z* where 2 = z-lBC is the number of collisions per second and Z* =P-l the number of collisions which are necessary to induce on the average one transition.On the other hand with a liquid we have to answer the question as to whether the same approximation is physically valid. The isolated binary collision model (IBC)1-6 invokes the rarity of a hard collision event inducing a transition whereas physical intuition about the molecular dynamics in liquids as well as a correlation analysis in liquids seems to substantiate the necessity of a more complex treatment extending to many-molecule correlations. This view is supported by the fact that virial co- efficients up to B5had to be introduced into equation-of-state calculations in order even roughly to describe pVT data of liquids.However the IBC approach has been surprisingly successful in correlating relaxation data with potential parameters and THOMAS DORFMULLER ET AL. models of the liquid The question then arises as to whether this success is due to a mere cancellation of approximation~,~*~* or whether it reflects the fact that the physical content of the IBC approach is a good approximation to the real state of affairs in a simple liquid. This question has been the subject of several investiga- tions but no conclusive answer is as yet available we believe that this is mainly due to the fact that the discussion was necessarily restricted to the few liquids for which reliable relaxation data were available either in a narrow temperature range or even at only one temperature.Although molecular vibrational relaxation data have been obtained mainly by means of ultrasonic methods the frequency range of this method is not appropriate for the study of most simple liquids. Hypersonic data as obtained from Brillouin scattering lie in the more convenient GHz range but are on the other hand less accurate. Brillouin scattering has hitherto not been used as a routine method over large temperature ranges. Our effort has been directed towards improving the accuracy of the Brillouin scattering technique and measuring a class of comparable molecules (CC14 CHC13 CsHSF C6H5Cl C6HI2) over a wide temperature range. The data obtained by applying this technique are relaxation strengths R(T) and relaxation times .r(T),both as functions of the temperature T.EXPERIMENTAL The optics and the apparatus used for the experiments in the present work are shown in fig. 1. The light scattering spectrometer consists of a single mode Ar ion laser L usually run at 200 mW on the 5145 A line. The incident beam is focused into a cylindrical cell S. The scattered light is observed at a given scattering angle with a spatial filter (pin holes Bl and lens Ll) a plane piezo scanned Fabry-Perot interferometer and a lens L3 focusing the central ring into a 0.1 mm pin hole. The photomultiplier output is amplified with an electrometer digitized and recorded simultaneously on a chart recorder and on a punch tape.Each spectral triplet is recorded with approximately 300 points. The spectrometer was operated with a finesse of 70-90 due attention having been paid to the divergence of the scattering angle and the stability of the interferometer. The scatter- ing angle could be varied from 20 to 160" by a system of 2 mirrors deflecting the incident beam. The temperature of the cell was controlled with a cold nitrogen stream and an electrical heater. The control thermometer TI was located inside the brass body M of the cell and the temperature was measured at T2 and inside the liquid at T3. M-FIG.1 .-Schematic diagram of experimental apparatus. (fi = 500 mm,fi = 300 mm = 500 mm pin hole Br = 1 mm diam. and Bz= 0.1 mm diam.) 108 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING The entire cell was located inside a stainless steel chamber with 18 plane windows situated in the scattering plane at different angles.The chamber could be evacuated for better thermal insulation at low temperatures. At temperatures above the normal boiling point of the liquid the chamber could be pressurized up to 5 atmospheres. LIGHT SCATTERING AND RELAXATION FORMALISM The light scattering spectrum in a relaxing medium can approximately be described by eqn (2) :11*12 The quantities appearing in this equation have the following interpretation The scattering wavevector k = (4nn/A)sin(8/2> The ratio of specific heats Y = CP/G The Rayleigh absorption rR = Kk2/poCp The classical non-relaxational absorption :To= (2~~13 + qv)k2/po The total absorption rB = To+ (r -1)/2rrV The relaxation parameter r = u2/ui The relaxation strength R = U~/U:.The sound velocity at the frequency o The sound velocity at the frequency 0 The sound velocity at the frequency 00 The heat conductivity The Brillouin shift u uo UOO K WB The scattering angle The mean density Index of refraction The vibrational heat capacity The heat capacity at constant volume The heat capacity at constant pressure Adiabatic relaxation time The physical content of the assumption upon which the approximation leading to eqn (2) is based is that the dissipative processes can be considered as small corrections to the elastic hydrodynamic response of the liquid to the dynamic variable. The spectrum expressed in this form contains the central Rayleigh line with the spectrum IR(k,m) the Brillouin stokes and antistokes lines with the spectra Ii(k,o) IA,(k,o) and the relaxation line with the spectrum ZM(k,m) z(k70) = zR(k,m) + zhl(k,m) + Iz(k,w) + I$(k,m) + rcorr(k,o)* (4) The last term represents an additional correction term taking into account the non- Lorentzian form of ZB(k,m).I3 For the present purpose we shall consider only the Brillouin doublet the Rayleigh line being irrelevant for relaxation.A fit of the experimental Brillouin lines gives us an experimental value of Fnand r thus permitting a calculation of rv and R via the relaxation equations THOMAS DORFMULLER ET AL. Additional information can be obtained from the experimental value of the Landau- Placzek ratio with f(t) = (r3-t2-r + R)/[r2-2t + R -y(t -l)'(R -t)]; I and IB are the integrated intensities of the central and the Brillouin lines respec- tively.One can use eqn (5) (6)and (7) to derive the quantities 7v,R,q,/q from the Brillouin spectra. Since however the uncertainty in I is very high and I has to be known with high accuracy to be of any value at all one either has to make some assumption about one of the variables thus reducing the number of variables to two I?"/% = 1 R = (C*-CMCP -YCJ or use more equations to determine the three unknown quantities. The most con- venient way to obtain more equations is to vary the wave vector k by varying the scattering angle 0. In doing this we obtain the frequency dependence of the sound velocity u(k) or of the relaxation parameter r(k).These data can then be fitted to eqn (5). A fit of the angle-dependent values of u(u) to the theoretical dispersion eqn (5) for CC14 at 20 "C is given as an example in fig. 2. The description of relaxation data is usually based on the Schwartz-Slawsky- Herzfeld theory (SSH) and the extensions introduced by Tanc~os.~ The basic ideas of this approach can be summarized as follows A linear head-on non-reactive collision model is applied where the molecular vibrations are considered to be harmonic and the rotational transitions are neglected. The T-V relaxation process takes place via collisional perturbation of the molecular states. This perturbation is essentially of the same nature in the liquid and in the gas (isolated binary collision or IBC approach).The molecular interaction enters the calculations through the repulsive inter- molecular exponential potential V(r,x) which is a function of the translational co-ordinate r and of the vibrational coordinate x. The r and x dependence of V(r,x) are separable as a product V(r)V(x). 1100 c I * 1000 E \ 6 900 1 10 100 -0xlO-'/ rad s-1 FIG.2.-Dispersion curve of the velocity of CCl at 20 "C. The resulting parameters are R = 1.370 7" = 51 PS. 110 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING The transition probability Pi + can be evaluated through the quantum mechani- cal expression for the perturbation integral Q =1tp 7V(r)tp,d.c whereiy and yj are the wave functions of the perturbed harmonic oscillator in the vibrational states i and j respectively.The solution of the Schrodinger equation of the system involves the use of the first-order disordered wave approximation. The average transition probability (Pi+j> per collision can be obtained by the integration over the Boltzmann-Maxwellian distribution of the relative collision velocities making the assumptions (,u/2)vY2 9 h and exp(4z2v/a*uF) 9 1. The SSH-Tanczos formula for (Pl -o) can then be given by the equation with In eqn (8) p is the reduced mass of the collision pair C the Sutherland constant ur the most effective initial relative velocity to induce a quantum jump with the energy hv Bo the potential well depth and a* a parameter in the used intermolecular potential V(r).To distinguish wave vector k from Boltzmann's constant the latter is printed kg. DISCUSSION OF THE RESULTS As already mentioned the experiments at an angle of go" when using eqn (6) enable us to calculate two of the three quantities R,zv and nv/vs. The temperature dependence of R,however is extremely helpful in obtaining additional information. Fig. 3 shows the temperature dependence of R compared with the theoretical Planck- Einstein results represented by two curves for each substance (indexed A-E). Curve (a)gives R for the whole of the internal vibrations and curve (6)gives R-for all but the vibration with the lowest frequency vmin. The experimental points have been calculated for CCl, and C6H5CI with the assumption vv/qs= 1 which is known to be approximately true for many Kneser liquids.For CC14 the points seem to fit the curve (a) and for C,H,CI they seem to fit curve (b). In order to obtain a fit of the experimental points for the other sub- stances we have to use different values for qv/qs. Table 1 gives the results for all the liquids which will be discussed here. Since the calculations of R and qv/qsdepend upon the absorption measurements given with an accuracy of lo%,the error in qv/qsis rather high; therefore the values are given without decimal figures. The decision however as to which vibrational levels have to be included in R is facilitated by the fact that any other choice than the one given here leads to quite unreasonable values for qv/qs(negative values for in- stance).One can see furthermore that this decision cannot be obtained from measurements at one temperature only at least with presently feasible experimental accuracies. The 4th column of table 1 shows the frequency of the lowest relaxing vibrational level which is the second lowest level in C6H5F C6H5CI and C6H12. The 5th column shows the isothermal relaxation times z at 25 "C. In the range of temperature where the relaxation process seems to have within THOMAS DORFMULLER ET AL. 1.5 r 1.1 2 1.0' 0 40 80 -Tloc 1 FIG.3.-Relaxation strength against temperature. Curve (1) gives R and curve (2) gives R-1. (a) CCL; (b)CHC13; (c) C~HSF;(d)C,H4C1 and (e) cyclohexane C6H12. the experimental error a single relaxation time z the temperature dependence of z can be described by eqn (8).In order to apply this equation we have to use the following assumptions (1) The IBC model is a good approximation for the situation leading [eqn (S)] to the relation (p1-0)1iquid = (pl-O)gas(l +-)e-"o/k~T. (9) C T The IBC approach has been subjected to numerous ~bjections.'~-'~ However these objections although theoretically sound could not disprove the validity of the IBC approach as a satisfactory approximation for the relaxation dynamics of liquids. It seems that at present the experimentalist does not have the choice of another approach. (2) The analytical form of the intermolecular potential V(r)is approximated by a Lennard-Jones potential 112 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING This potential is used mainly because of the availability of the parameters E and 0.This has to be fitted to an exponential repulsion potential V(r)= Voexp(-ar) + po at the classical turning point rc as the evaluation of the perturbation integral using the exponential form for V(r)gives an analytical form for Pl -o. TABLE 1.-vALUES OF THE RELAXATION STRENGTH R RATIO qv/qs,VIBRATIONAL MODE Itinst. AND ISOTHERMAL RELAXATION TIMES z FOR THE MEASURED LIQUIDS AT 25 "C substance R VVhS v1crn-l t"/PS CS2 R = 1.180 0 397 2640 CCl R = 1.378 1 218 124 CHCl3 R = 1.240 0 262 91 C6H5F R1 = 1.277 2 368 107 C6HsCl R1 = 1.217 1 296 58 CsH12 R1 = 1.297 2 384 63 a Calculated from the formula z = zv(Ry -l)/(y -1).It is well known that the Lennard-Jones potential especially its repulsive com- ponent is not satisfactory. In our case this is a serious drawback as T-V relaxation is mainly determined by the steepness of the repulsive potential at small values of the translational coordinate. On the other hand the virial coefficients are more depend- ent on the attractive form of the potential while the transport properties are sensitive to the repulsive part up to thermal energies. This means that Lennard-Jones para- meters evaluated from transport coefficients are to be preferred. Neglecting the attractive component we have to either use literature data for E Q rn or evaluate some of these quantities on the basis of other suppositions. We have used ccritvalues obtained from critical data.(3) The liquid structure can be described by means of a cell model. In connection with relaxation processes the models with movable walls either with an fcc structure (model A) or with a simple cubic structure C (model B) have been used quite success- fully. The resulting expressions for zBCare model A zBC= (21/6p0-1J3-o)/a model B:zBC = -a)/; where the average velocity 17is d = (8RT/nA4)'J2. The transition probabilities (PI -+ o) for the case of a harmonic oscillator can thus be obtained by using eqn (12) <Pl-+ 0) = %C/Z I1 -exP(-@v/T)l (12) and the calculated values of the isothermal relaxation times z. On the basis of the above mentioned three assumptions we have calculated the parameters m and 0for the liquid models A and B.The results are shown in table 2 Eqn (8) can be used to represent the results in a reduced form the reduction pertaining to the constants E 0 rn vinit for each substance. Fig. 4 represents a plot of log 2 against for various liquids at 25 "Cwhere 2 = (@o/~v)z~1~6exp(0v/2T) V~-o/((Pl,,)T1/6). Under the assumption that the choice of R and qv/qsis correct the plot should be linear and this result should confirm the interpretation of the relax- ation measurements as given in tables 1 and 2. This is indeed roughly the case for all THOMAS DORFMULLER ET .AL. TABLE Z.-LENNARD-JONES PARAMETERS CS2 A 434 23 4.82 4.44 4.5 1 B 22 4.22 CClJ A 335 25 5.38 5.88 5.79 B 24 4.83 CHCl A 427 22 5.17 5.43 5.04 B 20 4.5 1 C6H5F A 299 17 5.40 -5.99 B 15 4.69 CbHsCI A 346 17 5.51 -6.20 B 15 4.91 C6H12 A 329 24 5.69 6.09 6.04 B 21 5.05 substances but CS,.The point representing CS lies at a value of P which is smaller by an order of magnitude. The value of P given in the literature which was calculated applying a different is greater by a factor of 3. 6.5 I I 25 35 45 FIG.4.-A plot of log Zl against @i3. In conclusion one can say that the values of R,qv/qs,0 and m can be deduced from the hypersonic measurements. The conclusion however is not unique and has to be substantiated by plausibility arguments about the possible values of these quantities and about the validity of the liquid models chosen. This conclusion could be based upon the T-dependence of R and 7,which we believe is essential to show any discrepancies which arise from interpreting measurements at only one tempera- ture." We thank the Deutsche Forschungsgemeinschaft for support of this project.R. N. Schwartz Z. I. Slawsky and K. F. Herzfeld J. Chem. Phys. 1952 20 1591. R. N. Schwartz and K. F. Herzfeld J. Chem. Phys. 1954 22,767. F. I. Tanczos J. Chem. Phys. 1956 25,439. 'K. F. Herzfeld and T. A. Litovitz Absorption and Dispersion of Ultrasonic Waves (Academic Press London 1959) chap. 7 p. 260. 114 RELAXATION IN SIMPLE LIQUIDS BY POLARIZED LIGHT SCATTERING J. L. Stretton Transfer and Storage of Energy by Molecules ed. by G. M. Burnett and A. M. North (Wiley Interscience N.Y. 1969) vol. 2 chap. 1 p. 58. D. Rapp and T.Kassal Chem. Rev. 1969,69,61. T. A. Litovitz J. Chem. Phys. 1957 26 969. W. M. Madigosky and T. A. Litovitz J. Chem. Phys. 1961,34,489. M. Fixman J. Chem. Phys. 1961 34,369. lo R. Zwanzig J. Chem. Phys. 1961,34 1931. l1 R. D. Mountain J. Res. Nat. Bur. Stand. 1966 70A,207. l2 Th. Dorfmuller G. Fytas and W. Mersch Ber. Bunsenges. ghys. Chem. 1976 80 389. l3 C. Montrose V. Solovyev and T. A. Litovitz J. Acoust. Soc. Amer. 1968,43 117. P. K. Davis and I. Oppenheim J. Chem. Phys. 1972 57,505. lS J. Keizer J. Chem. Phys. 1974 61 1717. l6 H. Shin and J. Keizer Chem. Phys. Letters 1974 27 611. M. Sedlacek 2.Phys. 1975 ,4274,99. l8 R. Reid and T. Sherwood The Properties of Gases and Liquids (McGraw-Hill N.Y. 1966) p. 38. l9 J. Hirschfelder C. Curtiss and R. Bird Molecular Theory of Gases and Liquids (John Wiley N.Y. 1964) p. 1110.

ISSN:0301-5696    

DOI:10.1039/FS9771100106

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Memory Functions for Angular Motion in Liquids BY A. GERSCHEL Laboratoire de Physique de la Matikre Condensee Universite de Nice Parc Valrose 06034 Nice-France Received 1lth August 1976 Experimental memory functions are calculated corresponding to the orientational correlation function g(t) = <u(O) -u(t)) and to the rotational velocity correlation function gJt) = <i(O) -d(t)>. Analytical expressions are obtained corresponding to the Cole-Glarum and the Fatuzzo-Mason theories relating the spontaneous fluctuations of orientation to the frequency-dependent absorption. The properties of these functions are interpreted in the framework of generalized Langevin theory. Particular attention is given to the damped oscillatory features of the first memory function and to the slowly decaying features of the second.This analysisis achieved by investigating the temperature and density variations in some simple organic liquids (CH3F CH3CI CHF, CHCG and OCS). The occurrence of a plateau in the rotational velocity memory function is discussed. A tentative inter- pretation of this new feature is conducted by analogy with molecular dynamics experiments dealing with translational motion. 1. MOLECULAR RELAXATION IN LIQUIDS ANALYSED IN THE TIME DOMAIN REPRESENTATION Since it has been demonstrated by fluctuation-dissipation theory that current absorption spectra can be readily interpreted in the time domain thereby making accessible the molecular correlation functions this approach has been widely adopted for the analysis of elementary motions in liquids.Indeed spectral characteristics have been consigned to a second rank position because of the sudden appeal of correlation functions. Now after some years of rapid development it is useful to sort out the effective progress that has been made as well as to list the limitations of time domain analysis. As our contribution we aim to analyse the correlation func- tions themselves through their associated memory functions which appear to be very convenient tools for investigating the initial-time motions in relation to the overall dynamics. Molecular relaxation itself is investigated using a variety of experimental methods ; the results may be interpreted in the framework of more or less refined theories the predictions of which conform generally to the description of relaxation rates through one exponential decay time as remarked by Debye or through a combination of exponential decay times for more complicated liquid systems.At this stage only the Markovian nature of the process is characterized not the elementary steps which are responsible for this resultant law. The generality of Markovian behaviour is the ineluctable counterpart of the insensitivity of this law to the actual details of the motion.' This seriously limits the usefulness of relaxation studies for the evaluation of intermolecular potentials or of any genuinely microscopic (molecule-by-molecule) property characterizing the rates and mechanisms of the motion. In itself the derivation of the "microscopic " relaxation time is dependent on the approximations of conflicting theories2 which aim to account for the local reaction fields following 116 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS the molecular displacements.However the differences are generally quite small; certainly of the order of current experimental accuracy. Orientational correlation functions in so far as they depict the statistical overall tumbling motion of the constituent molecules usually show an exponential decay at long times. They may be considered as the interplay of two components auto-correlations reduced to monomolecular properties and crossed correlations from multimolecular interactions. Our concern here will be for the total functions only. Assuming now an analogy with Brownian motion valid if the interactions are frequent compared to the time scale of the motion one may consider the hydrodynamic friction coefficient to be a time-dependent property and thus further analyse this dependence.A general Langevin equation reads u(t) = -lKg(t -z)u(z)dz +f(t) or equivalently g(t) = -I$(t -z)g(z)dz. 0 In these relationships g(t) is a classical correlation function of the molecular property u(t),viz s(t) = (40) u(t)>. For the particular case of angular motion in polar liquids we consider u(t) to be a unit vector attached to the molecular axis supporting the dipole moment. The quantity f(t) has been termed a " random force "by analogy with the Langevin equation (actu- ally the vectorial field is a field of angular velocities); its correlation function is precisely the memory function which has already appeared in eqn (1.1) as a systematic retardation or "frictional " effect.These are well known general properties. They are discussed in the works of Zwanzig4 and M~ri,~ and further analysed by Harp and Berne6 and Pursuing our search for simple functions of the dynamics the structure of Kg(t) itself may be further analysed in terms of a second order memory function K:2)(t) defined by the integro-differential equation Only the computation of these functions for actual physical systems can determine whether the functions deserve attention. We now undertake this task for some selected simple organic liquids. 2. COMPUTATION OF EXPERIMENTAL MEMORY FUNCTIONS In what follows we use the notation 9g(t) =\me-i'utg(t)dt 0 to represent a Laplace transform and we write complex quantities as 9g(t) = RePg(t) -i ImZg(t) and E* = E' -id' (the dielectric constant).A. GERSCHEL 117 We will make use of the relati onship $4g(t) = im9g(t) -g(O) (2.2) and $4g(t) = -029g(t) -icog(0) -g(O) (2.3) where the odd derivatives at time zero e.g. the last term in (2.3) vanish for our classical molecular correlation functions and analytical intermolecular potentials. We have now to select a relationship linking the experimental data [here the values of ~'(o), ~"(m),eo and g(O) the Kirkwood-Frohlich correlation parameter (2.6)] with the molecular correlation function g(t). In a recent analysis3 we have shown that some discrepancies in calculating g(t) with different theories could arise with highly polar liquids which have a large zero-frequency value c0 and a very intense absorption in the FIR (far-infrared) and microwave frequency ranges.This does occur with some liquids considered here,9910 so for these we shall have recourse to the F-M expression (2.4) first established by Fatuzzo and Mason,ll and re-examined later by Zwanzig and Nee.12 Titulaer and Deutch2 indicate that the F-M analysis is preferred for liquids having a high static dielectric constant and intense absorption-dispersion features. For liquids of medium or low polarity13 the functions calculated using either F-M or C-G (Cole and Glarum) l4theories are indistinguishable; therefore we here adopt the simpler expression (2.5) of C-G which has the advantage of being linear in the complex quantity E* and which leads to a single relaxation time for an exponential decay of the correlation function.The expressions are with K-F do) = 9kTV where p is taken as the dipole moment in vacuum and the other symbols have their usual significance. Now we come back to eqn (l.l) which after taking the Laplace transforms yields which after combination with eqn (2.1) can be written Since by definition Re9.Kg(t) =LaKg(t)cosot dt by inversion for t > 0 one has K'(t) = Re9Kg(t)coscot do ikw 118 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS In this expression we insert the values of Regg(t) and ImZg(t) corresponding to eqn (2.4) or (2.5) as required that is F-M ReZg(t) = EOEtt[2(E12+ Ett2) + EW] (2.10) 0 (go -EW)(2EO + EW)(Et2 + Ett2) IrnZg(t) =g*){ 1 -EOIEt(E' -Ew)(2E' + Ew) + Et'2(2E' -Em)]} (2.11) 0 (Eo -Em)(2Eo + e,)(d2 + E"2) (2.12) (2.13) The resulting memory functions K,(t) possess very similar features to those of the functions grdf) = -g(t) = {d(O) u(t)> (2.14) the so-called rotational velocity correlation function~.'~ These in turn are compar- able under the same conditions with the angular velocity correlation functions.8 A systematic comparison of experimental functions KJt) and grv(t)has already been presented in ref.(3) where it was concluded that a slow decay of g(t) compared to grv(t) or the condition < (E~)~, are sufficient requirements for an acceptable identification.Turning now towards the second order memory function defined in (1.4) a very similar calculation applies in which one makes use of the analogue of (2.7) i.e. together with (2.8). This results in an expression for LPK(;)(t) similar to (2.8) the real part of which may be inverted thus leading to the analogue of (2.9) viz (2.15) with (2.16) According to Rahman,' such a second order memory function is the same as the memory function of the second derivative of g(t) defined in (2.14) provided a con- venient consistent definition of the memory function is used since gr,(t)is an " oscil-latory " function-i.e. of null integral 6grv(t)dt-whose memory function would otherwise incorporate a Heaviside step. Following the argument of ref.(7) such a well-behaved Krv(t)would be defined through We may now evaluate (2.15) by inserting the values of Re9Kg(co) and Im9Kg(co) computed from (2.8) and eqn (2.10) to (2.13). A. GERSCHEL A particularisation of (2.15) in the framework of C-G theory has already been obtained,ls it reads (2&0+ &')2 + &"2 o &t'2(2&(3 6" + Em)2 + [(2&,+ &')(&I -Em) + & ] cos cot do. (2.17) A somewhat more elaborate expression corresponded to the F-M relationship otherwise the analogue of (2.17). 3. PROPERTIES OF THE 1s-r AND ~ND ORDER EXPERIMENTAL MEMORY FUNCTIONS Taking advantage of very complete experimental information available for a few simple organic liquids at densities and temperatures ranging from near the triple point up to the vicinity of the critical point and all along the liquid-gas coexistence curve a systematic investigation of the properties of these functions is now possible.The selected liquids are at one extreme a triatomic linear molecule OCS and a symmetric top molecule CHCl,; these are medium polarity liquids. At the other extreme for highly polar liquids we have the three symmetric top molecules CH3F CH&l and CHF,. A fully detailed discussion of the experimental data is to be found in the appropriate referen~es.l~~l~*~~ Here we only specify very briefly the method employed and some correlative limitations. The experimental data consist of absorption-dispersion spectra over an extended range of frequencies.Zero-frequency determinations of the relative permittivity c0 and consequently of the K-F angular correlation factor g(O) are derived from classical capacitance measurement^.^ Dielectric relaxation studies involved deter- mination by waveguide techniques of both E'(CO) and ~"(o) at a few selected frequencies in the microwave range (namely 10 GHz 35GHz and 70 GHz). The fit of the data to Debye relationships was acceptable within the experimental accuracy. However a certain gap persists between the higher microwave frequencies and the lower fre- quencies of absorption spectroscopy in the FIR. Measurements are lacking between the wave numbers 3 cm-I on the lower side and 18 cm-l on the higher side though the higher frequencies in the FIR do range up to 180 cm-l.The correlative disper- sions in this range have been evaluated according to a distribution-of-resonances model.I7 For the time being spectral characteristics in the FIR provide the most complete and accurate source of information on short-time angular motions provided pure dipolar absorption is actually the main phenomenon responsible for the band profile. Limitations are mainly due to other than purely dipolar absorption features super- imposed on the raw experimental spectra in the FIR. A careful analysis of their origin is always necessary which implies that adequate corrections to the spectra are worked out before Fourier transformation into molecular correlation and memory functions. For the selected liquids having simple enough constituent molecules we obtained reasonably reliable frequency spectra of the neat dipolar absorption.Thus to begin with we reproduce in fig. 1 a comparison of &(t) with the cor- relation functions grv(t)defined in (2.14); perfect agreement is obtained close to the triple point but gradual departures develop as one moves towards expanded liquid 120 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS 0.5 0.3 0.2 0. 0. 0. 0. 0. FIG.1 .-The first memory functions K,(t) (open circles) compared with the rotational velocity correlation functions g,,(t). Both functions are in normalized representation. Three temperatures Tt are shown for liquid fluoroform (a) -156 "C (TR =-T-= 0,01) (6) -80 "C (TR= 0.42) Tc -Tt (c) and -3 "C (TR= 0.85).conditions. Similar conclusions are valid for the other liquids mentioned in this paper. The characteristics of the second order memory functions or rotational velocity memory functions K,,(t) are displayed in fig. 2 where they are compared with their associated correlation functions g,,(t). Striking features of KJt) are (i) an initial decay as sharp as for g,,(t) but certainly not sharper; (ii) a small amplitude oscillation rapidly dying out; (iii) an appreciable positive correlation effect persisting in the dense liquid for the time of damping of the librations [as represented by the oscillations of grv(t)] and thereafter falling off gradually; (iv) a progressive disappearance of such positive correlation as the dynamics evolve at the lower liquid densities obtained at higher temperatures.-08 0.7-06 - 05-0.5-0.5 -t 0.4 -$ 0.3-FIG.2.-The second memory function K,,(t) (upper curves) compared with the rotational velocity correlation functions. Same liquid as in fig. 1. (a) -156 "C (b) -80 "C (c) -3 "C. A. GERSCHEL The generality of these observations is made plain in fig. 3 and 4 where data over a set of temperatures are illustrated for each liquid. FIG.3.-The second memory functions normalized for two highly polar liquids (a)methyl fluoride at -140 "C TR = 0.01 (curve l) -80 "C TR = 0.32 (curve 2) and 0 "C TR= 0.76 (curve 3) and (b) methyl chloride at -97 "C TR = 0.01 (curve l) -60 "C TR = 0.15 (curve 2) and -3 "C TR = 0.40 (curve 3). -0.6 0.6-0 --I -0 20.5 --z k-0.L 0.3 0.3-0.2 0.1 0 0.1 [bl FIG.4.-Normalized memory functions KrY(f)for the two liquids OCS and CHC13.Carbonyl sulphide (a)is shown at temperatures -138 "C TR = 0.01 (l) -30 "C TR = 0.44 (2) and +90 "C TR= 0.90 (3). Chloroform (b) is at -60 "C TR = 0.01 (l) +25 "C TR = 0.27 (2) and +260 "C TR = 0.90 (3). In fig. 5 are illustrated the power spectra Re9Kr,(t) for the sake of comparison with the corresponding features of ~"(m)and Re9gr,(t) since for a better under- standing of the physical features we find it advisable to use both time-domain and frequency-domain representations. At low frequencies the behaviour mimics that of e"/m as is obvious from (2.17). The high frequency end of the spectrum on the other hand resembles an ~"(m) spectrum shifted towards higher frequencies.An additional shoulder can be noticed before the final decrease as a consequence of the rapid fall off of E" on the high frequency side of the FIR band [since this quantity contributes substantially to the denominator of the kernel in eqn (2.17)]. 122 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS For purposes of comparison the significant parts of the spectrum are therefore the high frequency characteristics of orientational relaxation including the region between microwave and FIR measurements (3-1 8 cm-l) together with the higher frequencies of the absorption in FIR. . From this observation we conclude that some features in the power spectrum of KrY(t)are less reliable than are the direct experi- mental spectra E"(CO) and ReA?grv(t)-the last being simply a(u) corrected for internal field effects.vlcm" v /cm" FIG.5.Normalized frequency spectra for two temperatures of liquid fluoroform left -140 "C TR= 0.10 right -3 "C,TR = 0.85. Curves (1) are the e"(i) representation. Curves (2) are the frequency spectra of the second memory function. Curves (3) are the frequency spectra of the rotational velocity correlation function-i.e. the absorption spectra a(i) as measured in FIR cor-rected for internal field effects. To clarify this situation we have simulated erroneous spectra within the limits of uncertainties in our experimental data with deliberate exaggeration of the distortion at the high frequency ends of both dielectric and FIR spectra.The distorted Krv(f) functions exhibit general behaviour quite similar to the original observations except that the amplitudes or elevation of the positive tails are shifted upwards or down- wards by amounts of 0.02 (in our normalized representation). Accordingly when the total amplitude above or below the axis exceeds 0.05 it may certainly be considered as significant. The steepness of the initial decay of Krv(t),as well as the period of the small damped oscillations is rather dependent on the decay of the FIR absorption equated at high frequencies and on the correlative dispersion. Finally the limit E~ to nIR2,suffers from being an estimate via the Lorentz-Lorenz relationship with measured polarizabilities.It would be much more satisfactory to have a true experimental determination of the refractive index variation and its high frequency limit in view of the presence of an (E' -term in the denominator of (2.17). To sum up a scrutiny of the uncertainties in the power spectrum leads one to be confident of the positive correlation features and their variations to within &2%,but at the present stage of our experimental knowledge quantitative assessments of the initial time decay of the function and the period of the superimposed oscillations can- not be made to better than &15% of the observed features. A. GERSCHEL 4. DISCUSSION Features of KJt) merit scant discussion since they are so close to the functions g,,(t) described in current FIR work.The oscillating tail is characteristic of strongly angular-dependent intermolecular potentials entailing correlating collisions the consecutive alternance being associated with reversal of the angular velocity within the local structure. As the density decreases the oscillations increase in period and become less pronounced evolving closer to the free rotation function yet still retaining some structure even in the vicinity of the critical temperature. Such evolution expresses in fact the weakening of the oscillator strength in the expanding structure. At long times the difference of behaviour between K,(t) and grV(t) lies in the fact that the first function maintains slightly higher (more positive) values especially at low densities. This is obvious from the properties where zD is the dielectric microscopic relaxation time.We now examine some newer aspects of the dynamics with the help of our findings in the Krv(t) representation. Considering fig. 2 to 4 it is evident that the important feature is the positive plateau extended over the time of damping out of the librations. This is the time that the molecule spends probing its environment before a local orientational rearrangement takes place. It may be termed a “ time of residence ” still keeping in mind that throughout this time the motion may be quite smooth or alternatively impulsive according to the details of the structural fluctuations. It is therefore some kind of generalized friction that is manifested by this plateau showing the effects of time-varying local torques imposed by the motion of the anisotropic surroundings about any definite molecule while the molecule performs correlated oscillations accompanying its reorientation.Such observations are reminiscent of those reported in earlier molecular dynamics work (computer simulation) dealing with translational velocities.ls Levesque and Verlet found a “ slowly varying part of the memory function positive for states not far from the triple point and negative if the temperature is substantially higher or the density lower ” interpreting the long-time part of this generalized coefficient “ as produced by an accompanying motion of the medium which according to the state reduces or enhances the self-motion ”. The authors taking advantage of their microscopic insight into the molecular motion were able to give a detailed interpreta- tion “In the general vicinity of the triple point where owing to strong cohesive effects the motion of the distinguished particle is on the average reversed after a short time the current gives rise to a retarded positive friction; that is the memory function is positive for large times.At higher temperatures or lower densities on the other hand where the initial motion of the distinguished particle tends to persist for large times the current pattern of the neighbouring particles enhances this forward motion. The memory function interpreted as a generalized friction coefficient is in that case negative for large times.” It is not unreasonable to transpose such a mechanism to rotational motion; our finding of a very similar behaviour of the angular memory functions does favour such an interpretation.It remains to be noted however that those currents revealed by our analysis of the rotational dynamics are possibly themselves subject to mutual interactions through some kind of dragging effect that shows up in the long-time tail of 124 MEMORY FUNCTIONS FOR ANGULAR MOTION IN LIQUIDS the memory functions K,,(t). Although a tentative explanation this is as much as we can say at present lacking as we do microscopic information as detailed as that provided by numerical molecular dynamics experiments. Furthermore one must realize that the negative portions sometimes encountered at the lower density range of the liquids were not sufficiently pronounced in the rotational dynamics studies to be certainly significant.Generally speaking we may conclude that if one reason for considering the second order memory functions is the search for simpler analytical objects than the rotational velocity correlation functions themselves one must admit that this expecta- tion is not met. More precisely any expected trend of successive memory functions towards a delta function is quite hopeless being in complete contradiction with experimental evidence. On the contrary the observed structure of the functions of higher order becomes increasingly involved and it is likely that the underlying physical processes so implied are correspondingly intricate in nature. In the meantime we have gained some physical evidence of the occurrence of coupling mechanisms between the fast and slow angular motions currently described as librations and reorientations respectively both kind of motions appearing as specialised aspects of the more general diffusion process.Furthermore we can ascertain that a detailed analysis of the motion does not reveal a white noise of molecular interactions indeed it has led us to propose the advent of some microscopic dynamic organization establishing local circular currents. Here memory functions have been a useful tool. They also have shown how general this structure is irrespective of the specific molecular shapes and polarities of some simple liquids though the structure is dependent on their physical state (density temperature).Finally it is also satisfactory that analogies have been revealed with the findings of molecular dynamics experiments for translational motion. R. I. Cukier and K. L. Lindenberg J. Chem. Phys. 1972,57 3427. U. M. Titulaer and J. M. Deutch J. Chem. Phys. 1974 60,1502. A. Gerschel C. Brot I. Dimicoli and A. Riou MoZ. Phys. 1977,33 527. R. Zwanzig J. Chem. Phys. 1960,33,1338 and in Lectures in Theoretical Physics (Interscience New York,1961) vol. 3 pp. 106-41. H. Mori Prog. Theor. Phys. 1965 33,423. G. D. Harp and B. J. Berne Phys. Rev. A 1970 2,975. A. Rahman StatisticaZ Mechanics New Concepts Problems Applications (University of Chicago Press Chicago 1972) p. 177. * P. Desplanques and E. Constant Compt. rend. 1971 272 1354.A. Gerschel MoZ. Phys. 1976,31,209. lo A. Gerschel I. Dimicoli J. Jaffre and A. Riou Mol. Phys. 1976 32,679. l1 E. Fatuzzo and P. R. Mason Proc. Phys. Soc. 1967 90,741. l2 T. W. Nee and R. Zwanzig J. Chem. Phys. 1970 52,6353. l3A. Gerschel I. Darmon and C. Brot MoZ. Phys. 1972 23 317. l4 R. H. Cole J. Chem. Phys. 1965 42 637. l5 A. Gerschel Comm. Phys. 1976 1 11 1. l6 I. Darmon A. Gerschel and C. Brot Chem. Phys. Letters 1971 9 454. N. E. Hill J. Phys. A 1969 2 398. l8 D. Levesque and L. Verlet Phys. Rev. A 1970,2 2514.

ISSN:0301-5696    

DOI:10.1039/FS9771100115

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Rotational Relaxation of Solute Molecules in Dense Noble Gases and the Relation with Local Anisotropy Fluctuations BY JAN VAN DER ELSKEN AND DAANFRENKEL Laboratory for Physical Chemistry University of Amsterdam Nieuwe Prinsengracht 126 Amsterdam The Netherlands Received 6th August 1976 Many molecular relaxation processes in fluids are sensitive to the time-dependence of local anisotropic density fluctuations. The role played by anisotropic density fluctuations in the rotational relaxation of a linear quantized rotor will be discussed in some detail. An expression for the dipole- correlation function of a probe molecule dissolved in a simple fluid will be derived along these lines. This expression will be shown to account for the experimentally observed density-dependence of the rotational linewidths of HCl in argon.Some simple models for anisotropic density fluctua- tions or more precisely for the four-point density-correlation functions will be discussed. Many different techniques can be used for the investigation of molecular relaxation processes. All these techniques provide us with macroscopic measurements ; hence the information which one may obtain from such measurements although commonly interpreted in terms of microscopic motions in the fluid is statistically averaged. Which statistically averaged property of the fluid one thus may obtain depends on the specific type of experiment. For instance inelastic neutron-scattering in simple fluids yields information on the dynamic pair-correlation function of the fluid whereas a phenomenon like collision-induced light-scattering probes the much more compli- cated time-dependent four-point density correlation function.In fact any micro- scopic process that is sensitive to the local anisotropic density fluctuations in the fluid is related to this latter correlation function. In particular the rotational relaxation of a probe molecule in a simple fluid depends on a density-correlation function of this type. A common feature in the description of these relaxation processes is that at a certain stage one needs to know the correlation function of a dynamical variable in terms of its dependence on the separation between two particles; for instance the collision-induced dipole in collision-induced dipolar absorption or the anisotropic potential acting on a probe molecule in the rotational relaxation process to be described below.Assuming this dynamical variable to be pairwise additive it is possible to relate the time dependent four-point density correlation function f(r, r,; r, r,; t) with the process under consideration. The experimental data depend only functionally onf(r, r,; r, r,; t) and this latter function cannot be obtained uniquely from the data. However different experiments often do yield complementary information about the four-point density correlation function. In particular depending on the type of experiment parts off(r, r,; r, r,; t) with different irreducible tensorial character may be probed. One may therefore construct models for the four-point density correlation function which if satisfactory should give good agreement with a number of experi- ments at a time.Although simple models forf(r, r,; r, r,; t) (and related cor-relation functions) have been constructed,1~2 no attempt has been made at least to ROTATIONAL RELAXATION OF SOLUTE MOLECULES our knowledge to combine information from different experiments in order to discriminate between different models. Such tests are badly needed as our knowledge of time-dependent higher-order density correlation functions is very limited to say the least. Even molecular dynamics calculations have been of little help mainly because tabulating let alone displaying a four-point correlation function is hardly feasible. In the following section we will investigate in some detail the rotational relaxation of a quantized linear rotor as an illustration of the role of higher order distribution functions in phenomena that are sensitive to the local anisotropy in the fluid.The discussion of other phenomena that depend on local anisotropy in fluids is quite similar. THEORETICAL DESCRIPTION In this section we will indicate how the shape of the far-infrared absorption spectrum of a linear probe molecule in a simple fluid may be related to the anisotropic density fluctuations in the host fluid around the probe. The interaction energy between a linear probe molecule and a perturbing atom at a distance r may quite generally be written as where ul(r)gives the r-dependence of that part of the molecule perturber potential that transforms as the I-th Legendre polynomial; I is the unit vector in the direction of the line joining the centres of mass of the molecule and the perturber; I is the unit vector along the molecular axis.If more than one atom at a time is perturbing the molecule the total interaction energy may (using the spherical harmonics addition the~rern)~ be written as Nco I where the pairwise additivity of intermolecular interactions has been assumed. In eqn (2) rl denotes the coordinate of the i-th perturbing atom; the origin of the co- ordinate system is fixed on the centre of mass of the probe molecule. Writing the number density at a distance r from the probe as p(r) =2N 6(r -ri) the above i-1 equation may be re-written as V(IJ = (4n/[21+ 111 m=2-I ylrn(l,)/bp(r)vl(r) Y:rn(lr)* (3) i=o Now if one assumes that the translational motions in the fluid are not sensitive to the orientation of the dissolved molecule the perturbation acting on the probe molecule has the character of a time-dependent external perturbation a 1 i=Q m-1V(1p; t) = 2 [4~/(2I+111 2 adt) Ydlp) (4) where a,,(t) is defined by alrn(t)=\*~(r; t)vl(r)yirn(1r)- (5) JAN VAN DER ELSKEN AND DAAN FRENKEL The time-dependence of the perturbation is obviously determined by the local density fluctuations around the probe ; the symmetry and range of the intermolecular poten- tial determines which type of density fluctuation couples most effectively with the rotational motion.Clearly as the molecule is assumed to be subject to an external time dependent perturbation the transitions induced in the probe through coupling with the density fluctuations will not obey detailed balance; in particular the per- turbation tends to heat the probe.In calculating the dipole correlation function of the probe molecule we will simply neglect this undesirable effect on the density matrix by assuming that the density matrix is still equal to the density matrix of the unperturbed rotor. Depending on the physical situation other approaches may be and in fact have been taken.4 The dipole correlation function (~(0).p(t)) =Tr[pp. U+(t)pU(t)] may be approximated by an expansion and partial resummation of the expression for the averaged operator product. The explicit expression for the dipole correlation function of the probe molecule then becomes where Pj,is the probability of finding a molecule with total angular momentum Ji (JtI ]Jf)is the reduced matrix element of the dipole-moment operator between states with total angular momentum Ji and Jf5and yjfji(t) is a function that depends on the correlation function of the perturbation acting on the probe molecule Apart from the 3-and 6-J symbols3 appearing in this expression yjfji(t)is only deter- mined by the Correlation functions gl(t),defined by The central quantity appearing in this expression is the density correlation function (p(r;O)p(r'; t)>.As this latter function is determined by the probability of finding a particle at distance r from the probe at time 0 and a particle at distance r' at time t it is determined by the four-point density correlation functionfir, r +r; r', rfP+r'; t)where rpand r' denote the coordinates of the probe at time 0 and t respectively.Integrating over r and rfPin the four-point correlation function yields the function {p(r; 0) p(/; t)> used above. The Legendre polynomials Pl(lr lf)determine which symmetry components of the density fluctuations around the probe contribute to the rotational relaxation. The distance over which anisotropic density fluctuations ROTATIONAL RELAXATION OF SOLUTE MOLECULES are felt by the probe molecule depends on the range of v,(r). For long times exp- [-pfji(t)] reduces to exp(-jfjit) with where G,(o) is the Fourier-Laplace transform of g,(t) G,(co) = e-'"'g,(t)dt.If all g,(t) are rapidly decaying functions of time (compared with the decay time of the dipole correlation function) one may replace exp[ -yjfjl(t)] by exp( -T'j lit) for all times. The power spectrum of the molecular dipole moment then reduces to a sum of Lorentzians with width Av+ = ReT'jfji and shift AV+~ = Im Tjfji. Eqn (9) indicates that there is a linear relation between the width and shift of all rotational lines and the Fourier-Laplace transform of the correlation function of the perturbing potential acting on the probe molecule. The widths and shifts only depend on the value of G,(cL)) calculated at frequencies corresponding to level spacings in the unperturbed rotor (including co = 0). In-tuitively this relation may be understood by noting that Re G,(cL))is the power spec- trum of the I-th perturbing potential.Simple first order perturbation theory would predict that the rate at which transitions occur from states J to J' is proportional to Gl(cojj,). If only the first one or two terms in the Legendre polynomial expansion of the molecule-perturber potential contribute to the rotational relaxation it is possible to obtain the corresponding frequency components G,(m) directly by inverting the set of linear eqn (9). The most interesting dynamical information may be obtained by using a probe molecule having rotational transitions throughout the frequency region characteristic for molecular motions. Rotational Raman line broadening may be analyzed in much the same way; comparison of the data obtained from far-infrared- and Raman-rotational line broadening of the same system would constitute a nice check on the method.At this stage it is useful to give the relation between local density fluctuations and a number of other relaxation processes as it shows how different experiments probe the density fluctuations over a different range. For collision-induced dipolar absorp- tion of a dilute solution of one noble gas in another the dipole correlation function may be written as where p(r; t) denotes the density at distance r from the dissolved atom; implicit is the assumption that collision-induced dipoles are pairwise additive. Collision-induced dipolar absorption gives information about the very short range density fluctuations as the collision-induced dipole decays to a good approximation exponen- JAN VAN DER ELSKEN AND DAAN FRENKEL tially with a characteristic length that is typically 0.1 times the Lennard-Jones diameter.Collision-induced depolarized light scattering occurs even in a pure fluid. The density correlation function that governs collision-induced light scattering is related to that four-point density correlation function which gives the probability of finding a pair of particles at roand ro+ r at time 0 and a pair (possibly the same) at rf0and r’o+ r’ at time t,f‘(ro;uo + r; rh; r; + r’; t). Averaging over roand r’o gives the density correlation function (p’(r; O)p’(r’; t)) which in contrast to the one defined before does not only contain information on the time-correlation of density fluctua- tions around a given particle but also on the correlation of fluctuations around different particles.The relevant correlation function for collision-induced light- scattering is \ p‘ <~’(r; o)P‘(~’;t>>f(r)f(r‘) PAIr where f(r) describes the r-dependence of the collision-induced anisotropic polaris- ability. For not-too-short distances this r-dependence is of course simply rW3. Clearly this is a much slower r-dependence than the one which determines collision- induced dipolar absorption. The long range part of the anisotropic potential between a linear molecule and a solvent atom is typically dominated by an r-6 or r7 term. Rotational relaxation measurements therefore probe density fluctuations over a range intermediate between the cases just mentioned.Very similar be it static information is contained in the r-dependence of g(r) through the relation Finally the isothermal pressure derivative of the van Hove self-correlation function that may be obtained from inelastic neutron scattering may also be expressed in terms of the four-point density correlation f~nction.~ RESULTS AND DISCUSSION Having indicated in what way rotational relaxation measurements are related to the local fluctuations in a fluid we now show how the description outlined above may be used to give a qualitative explanation of the density dependence of the width and shift of the far-infrared rotational lines of HCl dissolved as a probe molecule in argon.The far-infrared spectrum of HC1 in Ar has been measured by Frenkel et al.’ for argon densities between 100 and 480 Amagat at T= 162.5 K. It is observed that although the spectrum may be described by a sum of Lorentzians up to the highest density the density dependence of the different linewidths is distinctly non- linear and moreover different for different lines. This effect is in striking contrast with what is usually observed in dilute gas systems. It demonstrates the need for an approach that takes fully into account the dynamic particulars of higher density systems. To compute the rotational line-widths one needs to know the correlation functions of the different irreducible parts of the perturbing potential gl(t). These correlation functions depend on the shape of ul(r) and of course on the dynamics of the medium.As there hardly exists any a priori knowledge of the correlation functions gl(t),we undertook a molecular dynamics (M.D.) study to compute gl(t)for I = 1 and I = L9 The M.D. calculations were done using the Lennard-Jones parameters Elk = 119.5 K ROTATIONAL RELAXATION OF SOLUTE MOLECULES and o= 3.405 A for argon. As the mass of HCl and the Lennard-Jones o of the HCl-Ar potential are rather close to the corresponding argon values HCl was replaced in the computation by argon in order to improve the statistics of the M.D. calculation. For simplicity ul(r) and uz(r)were assumed to have the same r-dependence as u&); the strength of the anisotropic potential was used as an adjustable parameter.Fig. 1 to 4 show the correlation functions gl(t) and gz(t)and their Fourier transforms at 0 I I 1 I 5 10 15 t 10-13s FIG.1.-The correlation function of the part of the anisotropic perturbation that transforms as an irreducible tensor of rank one. The correlation functions were computed for argon densities of 100(1) 300(2) 500(3) and 784(4) Amagat at T=160 K. All functions are drawn to the same arbitrary scale. densities between 100 and 784 Amagat. From these data the linewidths of a number of rotational lines of HCl in argon were calculated. In table 1 these computed widths are compared with the experimental values. The computed linewidths clearly reproduce the non-linear density dependence that is. observed experimentally.The consideration of the simultaneous perturbation by many particles seems there- fore to be a fruitful approach. Unfortunately the method is restricted to systems of which information about the many-particle dynamics is known inferred from M.D. calculations or otherwise. It is of course of considerable interest to compare the dynamics of local density fluctuations in different fluid systems. One simple way to extrapolate the results of the M.D. calculations to other fluids would be to scale the results of the computations to the mass and L.J. parameters of the fluid under consideration. Such an extra- polation would however also require the mass and L.J. parameters of the probe to be scaled in the same ratio which is physically quite unrealistic.Indeed simple scaling of the argon results to krypton and xenon does not seem to reproduce the experimentally observed increase in rotational structure of the dissolved HCl with JAN VAN DER ELSKEN AND DAAN FRENKEL FIG.2.-The correlation function of the part of the anisotropic perturbation that transforms as an irreducible tensor of rank two. The correlation functions were computed for argon densities of 100(1) 300(2) 500(3) and 784(4) Amagat at T = 160 K. All functions are drawn to the same arbitrary scale. the mass of the solvent noble gas atoms.1° This indicates not surprisingly that the ratios of mass and size of the probe to the mass and size of the solvent atoms plays an important role in determining the dynamics of the solvent around the probe.A second important point to note is that if one is studying molecular fluids it is not just the translational motion but also the internal motion (rotation) of the solvent molecules that modulates local density fluctuations. For instance the L-J para-meters and mass of SFs and CC14 are quite close yet HC1 dissolved in the former liquid does show rotational fine structure in the far-infrared absorption spectrum l1 whereas dissolved in the latter surrounded by clearly less spherical molecules it does not.12 To get some physical insight into the factors which determine local density fluctuations in simple fluids we now discuss a few simple models that describe limiting cases of the microscopic dynamics of the fluid. Simplest is of course the very dilute gas.In the dilute gas only binary encounters contribute to the local density fluctuations. The correlation functions gI(t)are therefore only dependent on the dynamics of a pair of particles during an encounter. In lowest order increas- ing the density only increases the number of encounters thereby affecting the magni- tude of gI(t)but not its shape. As can be seen from fig. 1 and 2 increasing the " argon " density in the M.D. experiment from 100 to 300 Amagat clearly changes the shape of gl(t)and gz(t),thus indicating that ternary and higher order encounters ROTATIONAL RELAXATION OF SOLUTE MOLECULES I 0 100 200 u /cm-’ FIG.3.-The Fourier transform of gl(t) at T= 160 K as a function of the argon density. The functions G,(o) for 100(1) 300(2) 500(3) and 784(4) Amagat are all drawn to the same arbitrary scale.become important. (A correlation in density fluctuations during successive en- counters is actually present even in the dilute gas giving rise to the well-known intercollisional interference dip in collision-induced dipolar absorption,13 but this effect only adds a small be it slowly decaying part to the relevant correlation func- tions). A suitable model for the high density limit is the ideal harmonic solid. Density fluctuations around a given point in a solid may be analysed in terms of phonons. For simplicity we assume the solid to be an elastic continuum. Although the probe molecule takes part in the collective motions of the solid we neglect this aspect of the problem and assume the probe to be fixed.Using the expansion of a plane wave in spherical waves,3 it may be shown that the Fourier-transform of gl(t) is proportional to D(m)Vf(m)where D(co)is the density of states of longitudinal pho- nons and Vf(m) is defined by dk/ /6[0 -~(k)] V:(cu) = V$(k)G[co -~(k)] dk (13) where Vl(k) = Lrnv,(r)j,(kr)4nr2dr. j,(kr)is the I-th spherical Bessel function vl(r) is the r-dependent part of the aniso- tropic potential and a is the radius of the probe molecule. It is important to note that this (oversimplified) model indicates that the power spectrum of the anisotropic JAN VAN DER ELSKEN AND DAAN FRENKEL u/cm-’ FIG.4.-The Fourier transform of gz(t) at T = 160 K as a function of the argon density. The functions Gi(o) for 100(1) 300(2) 500(3) and 784(4) Amagat are all drawn to the same arbitrary scale.TABLEDENSITY DEPENDENCE OF ROTATIONAL LINEWIDTHSOF HCl IN ARGON initial J Av112/cm-1 p = 100 Amagat p = 300 Amagat p = 480 Amagat exp. calc. exp. calc. exp. calc. J=4 2.66 f0.1 3.31 7.51 0.24 7.51 12.0 f1.8 12.01 J= 5 2.09 f0.22 2.28 5.67 f0.37 5.20 8.01 f0.24 7.81 J= 6 1.65 f0.16 1.54 3.91 f0.24 3.66 5.58 f0.3 5.04 J= 7 1.21 & 0.19 1.09 2.46 f 0.36 2.41 3.23 f0.16 3.36 J= 8 0.87 f 0.06 0.76 1.54 f0.32 1.55 1.88 rrt 0.17 2.36 potential is proportional to the frequency-distribution of the solvent. It is tempting to consider the strong increase with density of gl(co) in the frequency range between 30 and 100 cm-l as a reflection of the fact that in the dense fluid propagating modes in this frequency range start to appear.If the fluid were an elastic continuum D(m) and therefore g,(cu) would be zero at cu = 0; the fact that the computed gl(co) are clearly distinct from zero at cu = 0 is an indication that the collective excitations that are responsible for local density fluctuations have a finite lifetime. The picture given above may be extended in such a way that it does take the decay of density fluctuations into account. Instead of assuming that the auto- ROTATIONAL RELAXATION OF SOLUTE MOLECULES correlation function of a density fluctuation with wave-vector k is simply exp[-ico(k)t] one may get a more realistic picture by putting N-' <p-k(O)pk(t)) = I(k t) (15) where I(k,t) is the intermediate scattering function; this latter function may be obtained experimentally from inelastic neutron scattering.One then obtains (16) Although the simple models described above give a qualitative insight into the factors that influence local density fluctuations in a fluid they are very crude and only suitable to make order-of-magnitude estimates. More sophisticated models are needed to describe local anisotropy in a fluid in a realistic way. One interesting possibility would be to make an approximation for the four-point correlation function similar to the superposition approximation for g3 f(rp;rp + r; ri r; + r'; t) = pg(r)g(r')Gs(ri -rp; t)G(r' -r; t). (17) Such an approximation may be tested directly through a molecular dynamics simu- lation.The attractive feature of this approximation is that it yields expressions for all the relevant correlation functions described above without adjustable parameters. For instance in this approximation gl(t) may be written as gl(t) = p / v:(k)I,(k t)I(k t)4nk2dk (18) where v,(k) = ~mg(r)vl(r)jl(kr)4m2 dr; where I,(k,t) is the self-part of the intermediate scattering function. In the correla- tion functions for collision induced light scattering I,(k,t) is replaced by I(k,t). The correlation function for collision induced light scattering should always be positive in this approximation as the integrand is always positive. M.D. calculations by Alder Strauss and Weiss do indeed show this behavi0~r.I~ However possible deviations from this approximation are in fact most interesting as they would indicate that the study of time-dependent anisotropic density fluctua- tions does yield new information that is not contained in the dynamic pair correlation functions.This work is part of the research of the Foundation for Fundamental Research of Matter (F.O.M.) and was made possible by financial support from the Netherlands Organisation for Pure Research (Z.W.O.). M. Gillan and J. Woods Halley Phys. Rev. A 1971,4,684. * H. C. Torrey Phys. Rev. 1953,92 962. A. R. Edmonds Angular Momentum In Quantum Mechanics (Princeton University Press Princeton N.J. 1957). ' D. Robert and L. Galatry J. Chem. Phys. 1971,55,2347. The reduced matrix element (JITl(k)llJ'> used in the text is defined by <JmIT(kq)lJ'm'>= (-l)J-m( -; ) ;, <JIIT(k)llJ'> (2J + 113.S. A. Rice and P. Gray The Statistical Mechanics OfSimple Fluids (Interscience N.Y.,1965). JAN VAN DER ELSKEN AND DAAN FRENKEL 'P. Egelstaff C. G. Gray and K. E. Gubbins Phys. Letters 1971 374321. D. Frenkel D. J. Gravesteyn and J. van der Elsken Chem. Phys. Letters 1976,40,9. D. Frenkel and J. van der Elsken Chem. Phys. Letters 1976,40 14. lo R. M. van Aalst and J. van der Elsken Chem. Phys. Letters 1973 23 198. G. Birnbaum and W. Ho Chem. Phys. Letters 1970 5,334. l2 P. Datta and G. M. Barrow J. Chem. Phys. 1965,43,2137. l3 J. van Kranendonk Canad.J. Phys. 1968,46,1173. l4 B. J. Alder H. L. Straws and J. J. Weis J. Chem. Phys. 1973 59 1002.

ISSN:0301-5696    

DOI:10.1039/FS9771100125

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Studies of Ion-ion and Ion-molecule Interactions using Far-infrared Interferometry BY COLIN BARKER AND JACK YARWOOD* Department of Chemistry University of Durham Durham City DHl 3LE Received 1I th August 1976 The far-infrared spectra of tetra-n-butylammonium halides in benzene chloroform and carbon tetrachloride are interpreted with the aid of a dynamic model based on the stochastically modulated oscillator theory of Kubo. Although the model is not necessarily unique for this particular (and com- plicated) system it does enable a reasonable interpretation of the observed band frequencies widths and intensities and their variation (or otherwise) with changes of salt solvent and temperature. All the data are consistent with situations in which the ion-pair (or aggregate) vibrations are stochastically but relatively slowly modulated by interaction with the surrounding solvent molecules The perturbation of the solvent molecules mainly by dipole-induced dipole interaction is shown to be severe.There is also evidence of strong coupling of the solvent "collision mode " and the ion-pair vibration and that the latter is strongly overdamped by a high Langevin friction constant. The implications of this work for the interpretation of data obtained at lower frequencies are considered to centre on the large solvent-solute interaction and this large microscopic viscosity coefficient. The far-infrared (submillimetre) absorption by electrolyte solutions has been examined both directly 1-3 and indirectly 43 (using high frequency microwave measure- ments).Recent work in our laboratory has been aimed at a comprehensive quanti- tative study of a range of tetra-alkylammonium salts in (supposedly) poorly solvating media in an effort to throw further light on the nature and origin of these absorptions. In this paper we examine the possibility of constructing a dynamic model for the ion- pairs (or aggregates) which will allow us to interpret the experimental data in a meaningful way and which will (hopefully) provide further insight into the vibrational and relaxational behaviour of the system. We also consider here the ways in which our model may be developed and tested and the similarities and differences between this and other models-particularly ones used to interpret ''dielectric " (microwave) data.49 5.7-12 EXPERIMENTAL The measurements were made using a Beckman-R.I.I.C.Ltd. FS720 interferometer and fixed pathlength cells of stainless steel with high density polyethylene windows. A Beckman-R.I.I.C. Ltd. variable temperature cell holder (VLT-2) and temperature controller were used for measurements between 285 and 350 K. The tetra-n-butylammonium salts were pur- chased from Eastman-Kodak Ltd. They were dried (when necessary) and the water content of the solutions was monitored by the Karl-Fischer method. The solvents were either " AnalaR " or " Spectro " grades and were dried over molecular sieves immediately prior to use. The spectra shown here are ratioed absorbance spectra (corrected for gain differences) of the appropriate solution against pure solvent background.The spectral resolution is -2.5 cm-'. Our methods of estimating the precision of the data of constructing a meaning- COLIN BARKER AND JACK YARWOOD ful " base line " (for intensity measurements) and of checking on the effects of small amounts of water in the solutions have been published previ~usly.~ RESULTS Typical spectra of Bu,"N+Cl' in the three solvents used and of Bu,"N+Br- in benzene are shown in fig. 1-4 together with the decomposition (for Bu,"N+CI- solutions) into their separate component^.^ The resulting band frequencies half- widths and intensities are given in table 1. Our initial interpretations of these spectra have already been published3 but since any model which is constructed must at least be consistent with the experimental data it is appropriate here to summarise the principal features of the observed spectra.(a) The band system (for Bu,"N+Cl- solutions at least) can be decomposed into three separate bands (called A B and C for convenience) each of which appears to be consistent (both statistically and phenomenologically) with the (assumed) Gaussian pr0fi1e.l~ (b) The three bands are of distinctly different widths and intensities and these parameters are to some degree (see discussion below) dependent on variables such as salt concentration and temperature. However the main features of the spectra remain essentially unchanged for a given anion (but change when the anion is changed). In particular the spectra are to a large extent solvent independent.(c) The assignments5 of bands A and B to phenomena associated with solvent (v,) and ion-aggregate (vCA) (ion-pair in dilute solution) respectively is fairly straight- forward. Thus v corresponds to a (perturbed) collision model4 of the non-polar solvent while v, is taken as a vibration of anion against cation in an ionic aggregate- whose exact nature in concentrated solutions is somewhat obscure. The assignment of band C is more difficult but we shall see that our model points to a reasonable interpretation in terms of a combination band of vs + v, (see below). THEORETICAL CONSIDERATIONS Previous models5*7-9 relating to the relaxation of a system of solvated ions have been aimed at interpreting data c011ected~~~~~-~~ at much lower frequencies (typically between 100 Hz and 300 GHz i.e.up to 10 cm-l) on the dipolar relaxation of the perturbed solvent and of the ion-pairs (or aggregates of ion-pairs). As pointed out by Lestrade et aZ.,5 the relaxation (or other) phenomena which are observed depend on the frequency of the radiation used in the experiment and so in the far-infrared region (at frequencies corresponding to > 1000 GHz) we expect to " see '' phenomena which are characterised by relaxation times in the 0.1-1 .O ps region. Nevertheless it is valid to observe that microwave measurements lead to relaxation times for the ionic aggregate lifetime (7,in Lestrade's notation)7 and for the elapsed time between ionic collisions (z,) of the order of 100-300 ps.597-9 This means that as far as the far-infrared region is concerned the ion-pairs can be treated as though they are quasi- stationary.This is one of the assumptions inherent in the treatment given below. The other major assumption is that the vibrational mode considered is that of a solvent-surrounded ion-pair (see fig. 5). Except in very dilute solution this is unlikely to be realistic of course but we note that the spectra vary little over the concentration range studied down to 0.05 mol dme3. Further we have now studied (with the aid of a polarising optical system and a cryogenic detector)6*15 some of these systems down below 0.01 mol dm-3 without any major spectral changes. So there may be some empirical justification for considering an ion-pair in pseudo-isolation.138 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS -D wavenumber/crn'l RG.1.-Far-infrared Spectrum of Bu4*N+C1- in benzene. X observed spectrum; A vs band at -65 cm-'; ByvcA band at -115 cm-'; C vS + v,A band at -185 cm-'; D total computed band envelope. wovenumber / cm" FIG. 2.F-r-infrared spectrum of Bu4"N+C1' in carbon tetrachloride. X observed spectrum; A v band at -70 cm-'; B vCAband at -120 cm-'; C v + vcA at -185 cm-'; D total computed band envelope. COLIN BARKER AND JACK YARWOOD w avenu mber/ cm-1 FIG.3.-Far-infrared spectrum of Bu4"N+C1- in chloroform. X,observed spectrum; A v band at -70 an-'; B v,A band at -120 cm-I; C,v + vcA band at -180 cm-*; D total computed band envelope. w avenumber/cm-l FIG.4.-Far-infrared spectrum of Bu4"N+Br-in benzene.Bands A and B are now very much closer together and absorption in 150 cm-l region is very small. The observed spectra and initial interpretation3 (see above) point to the need to consider a stable vibrating ion-pair (with a lifetime which is long compared with the vibrationalperiod) whose solvation shell of surrounding solvent molecules is consider- ably perturbed by interactions with the charged species. Consider (fig. 5) a representative solvent molecule (S) at radial distance R from the ion-pair CA (of finite length rCJ. A multipole expansion16 of the potential at S due to the ion-pair is TABLE1.-FAR-INFRARED SPECTRAL PARAMETERS OF Bu4"N+X-SOLUTIONS IN ORGANIC SOLVENTS band "A " (v,) band "B " (vcA) band "C " (vs + vca) salt solvent temp./K iimax/cm-' Av'i/ern-' in tensity" ijmax/Cm-' Ai+/cm-l intensi t yo iim,x/cm-l AiiJcm-l intensitya Bu,"N+Cl-C6& 285-305 75 f3 56 f 2 2400 118 4 2 52f 3 3800 182f 2 60f 10 -1400 305-328 73 f2 58 f2 2600 117 f2 54f 2 3900 181 f2 54f 5 -1200 328-350 75 f5 62rt 3 3300 118 & 2 52f 5 4200 180f 3 55% 5 -1500 293 73 f4' 724.8' 6300' 118 & 3' 53 f 4' 58Wb 181 f6b 724 19' ~3500~ ~ ~~~~~~~~ + Bu4"N C1-CC4 293 75 f 6 64f 6 3700 114f 2 50f 2 7200 184f 3 80f 15 2500 Bu"~N+ C1-CHC13 293 70f 7 50f 11 1800 116f 2 52f4 7500 175f 7 84f 16 1500 & 0 r C C Bu4"N+Br-C6H6 293 65 f 4 75 4 3300 78 f2 56 f3 1900 --C m 0 C C C Bu4"N+Br-CHCl 293 65 & 5 -3900d 80 f5 -3900d --C r m Intensities have -&1O-15% error in all cases.They are based on total solvent (band " A ") or total solute (bands "B " and "C ") concentrations. The units are dm3 mol-I cm-2. Data at 293 K are averages over whole range of concentration from 0.05 to 0.8 mol dm-3. Variable temperature data at 0.25 mol dm-3. Band not intense enough to measure. Total band intensity of usand oCA. COLIN BARKER AND JACK YARWOOD 141 where Q is the electronic charge. If R & rcAthen the 1st term (dipole term) in (1) is dominant and it will be a reasonable approximation to consider the solute-solvent interaction to be controlled mainly by dipole-induced dipole interactions. For a polarisable ion the dipole moment is,14 and for the tetrabutylammonium halides pcA is expe~ted~~pl~ to be about 30 -36 x 10-30Cm.Thus the dipole induced in a typical solvent molecule pg = a,E (3) [where the field strength is given'* by E = -grad (V(R,O)}] gives rise to a potential energy of interaction Udp-indp = -*O!,E2 (4) FIG,5.-The ion-pair and surrounding solvent molecules. Parameters needed for the calculation of the dipole induced in a representative solvent molecule by a point dipole pcA. where CC,is the polarisability of the solvent molecule. It should be noted that for benzene the polarisability is distinctly anisotr~pic,~~*~~ and this may result in orienta- tional effects around an ion-pair. The results of applying eqn (3) and (4) will of course be to calculate significant dipole induced interactions between the ion-pair and solvent.We now have to note that due to the motions of the solvent molecules and also due to the vibration of the ion-pair the dipole-induced dipole interaction of eqn (4) is time dependent. In particular it will depend on rcA(t)and R(t) and since in the simplest case (2) reduces to pcA= Qr,, it then follows from (l) (3) and (4) that Udpindpis proportional to the square of the total ion-pair-solvent distance [see ref. (18) p. 1261. However if R(t)is taken to be a slowly varying function of t (for the simplest case) then we can write Udp-indp = k(l)rz.4-(5) (However see discussion section for details of possible deviation from this simplified expression for U.) k(t) then represents the time dependence of the interaction potential and is a complicated function which depends on (i) the relative orientations and separation of the solvent and ion-pair and (ii) the (possibly anisotropic) l9polaris-ability of the solvent molecules.Thus as k(t) fluctuates the ion-pair vibration suffers a stochastic modulation which will of course in principle affect the spectrum. The ion-pair vibration may now be treated using the general formalism developed by Kubo20 for calculating the spectral line shape (or associated relaxation function) of a randomly modulated oscillator. (This treatment has recently been successfully applied2lVz2to the case of a weak hydrogen-bond vibration in a " bath " of surround-ing solvent molecules.) 142 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS The Hamiltonian for the vibration of the ion-pair is written as =$&A/m + 3m rZA ($A + kr:A =$&$m 3-$m rgA (WgA + 2kfm) where m is the reduced mass givenB for a contact ion-pair as the subscripts c A and s referring to cation anion and solvent respectively.We may now regard the operator (6)as a stochastic operator written as x(t) = *p:$m + 4m rgA m2(t) where the effective time dependent oscillator frequency m(t) is obviously given by Thus o(t) = mo + ml(t) where wo= oCA and ml(t) =k(t)/mw, in our case. The transition dipole relaxation function p(t) of the:oscillator is given 2o by p(t) = (exp (iP(t')dtf)) (9) 0 and is easily calculated from the spectral distribution I(m) since the transition dipole moment autocorrelation function is and J -ca Although the calculation of q(t) is easy (see fig.6) in order to proceed further with the interpretation in terms of the molecular properties of the system it is necessary to know the form and distribution of the stochastic process k(t),ofeqn (8). Although expressions for k(t) are complicated (and we do not believe that with the present quality of spectral data band fitting is justified) we do have some experimental evidence as to the probable source and properties of this process. We first note that the observed spectral distribution of rcA and its associated p(t) (fig. 6) are well-approximated by Gaussian functions with relaxation times of about 0.4 ps. In the slow modulation limit20*22 [i.e.,for a k(t) process which is slow com- pared with the rate of decay of the dipole moment relaxation function p(t)] the spectral distribution follows the statistical distribution of k(t)-which would in that case be Gaussian.Further we note that the band profile is expected22 to be Gaussian (by the central limit if the number of near neighbour solvent molecules is large. Since it is unlikely that benzene molecules are rigidly "bound "to the ion-pair in this case it is doubtful if the solvation number means very much for these solutions. However the tetralkylammonium ions are known 26 to have effective ionic sizes (without solvation shell) of 4-5 A and the number of benzene near neigh- bours is estimated to be about 20-30. However k(t) is clearly not a stationary COLIN BARKER AND JACK YARWOOD timeips FIG.6.-Relaxation functions p(t) of the v6 and vcA vibrations in the Gaussian approximation.process. In the liquid phase we know that the solvent molecules collide with the ion- pair (or aggregates) and with each other. Indeed we see the effects of these collisions in the "A " band of the observed spectrum. So one component of k(t)will fluctuate at the solvent collision rate and another component will fluctuate at the solvent-ion collision rate. Since the motion of the ions is rather slow it is principally these motions of the solvent molecules which lead to the variation of k with time. Thus the collision mode of the solvent may be treated as a stochastic process. If the change in R caused by collisions is AR then R(t) = Ro + AR(t) (12) and since the effect on mcAis through dipole-induced dipole 16*18interactions k(t) = C'R-6 = C'(Ro + AR)-6.(13) Expansion of (13) gives terms in AR AR2 etc. which when used to modify the Hamiltonian will lead to terms in &*AR rEA AR2 etc. These will lead in principle to coupling of the collisional (v,) and ion-pair (vCA)modes. The resulting effect which is expected to be large because v and vcA are very similar in frequency is that a band at zFs + GcA is predicted. The presence of a band very close to 9 + tcAin our spectra-a band without any other obvious explanation-provides strong support for the general validity of our model and the preceding theoretical considerations. 144 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS DISCUSSION Clearly the principal observed spectral features are adequately explained with the aid of the model of the electrolyte/solvent system which is outlined above.This model is based on a stochastically modulated ion-pair vibration the modulation being provided by the surrounding solvent molecules which are themselves severely per- turbed by the effects of dipole-induced dipole interactions. Nevertheless it is useful to consider the data in rather more detail in order to investigate more closely the nature of the molecular processes involved in giving rise to these spectra. Consider first the "A " band (vJ. The results in benzene solution show that both the width of this band and its intensity increase with increasing temperature while the band centre remains fairly constant (with a slight tendency to show a high frequency shift but the scatter of data is too high for this tendency to be confirmed).These results are of course what is expected if the band is due to collisions of the solvent molecules. For example Pardoe 27 found that collision bands of non-polar solvents increase slowly with an increase in temperature (in contrast to the behaviour of the "libra-tional" band27-29 of a polar solvent which decreases rapidly in frequency as the temperature is increased). Thus our data strongly support the idea that band "A " is caused by solvent molecules translating against one another (and against the ions in solution) the large perturbation being due to the greatly enhanced fluctuating electrical fields caused by changes in the solvent-ion distances.Although any solvent might be expected to show this effect a correlation is expected between the size of these fluctuating fields (and hence spectral intensity) and the polarisability. It is clear from table 1 that indeed the largest "A " band intensity occurs for benzene which is known to have high p~larisability.~~*~~ (Since carbon tetrachloride also has a high p~larisability~l it appears that the anisotropy of the polarisability of benzene does play an important role.) The data for carbon tetrachloride and chloroform show some interesting features in that the v band frequency is virtually solvent independent. Carbon tetrachloride a collisional band at 44 cm-l in the pure liquid while chloroform an absorption maximum at -35 cm-l.Of course the situation is complicated in chloroform by the presence of permanent solvent dipoles and the p~ssibility~~ of specific solvation of the anion but it appears that interaction with the ion-pairs (of aggregates) in solution is so strong that this produces virtually the same "effective " collisional frequency of the perturbed solvent molecules. This may be related to the change in vis~osity~~*~~ of the medium on going from solvent to solution (some of the solutions are highly viscous34 especially at the higher concentrations) but in that case it is surprising that benzene does not show the same effect. The results in benzene show that this band is strongly anion dependent the frequency and intensity of the band decreasing somewhat on going from CI-to Br- (table 1).This implies that the effects of ion-solvent interaction are smaller for the bromide salt. Such effects depend of course on the " effective " dipole moment of the ion- pair [pCAof eqn (2)] and some such values have been calculated from dielectric data.12J7*35 Since the effective interionic distance is expected to be greater for the bromide salt the effective dipole moment should be greater. Comparison of the observed dipole moment data33 for the corresponding Bu,"NH+X- (X = C1 Br I) salts shows that this is indeed the case. However the exact nature of the species present (and their electrical properties) depends on the extent of ionic aggregation so it is difficult at this stage to be sure of the effects to be expected.Measurements of the far-infrared spectra at much lower concentrations l5 should throw further light on this problem (and remove possible effects due to gross viscosity changes). Considering now the " B " band (designated vcA) we see from table 1 that the COLIN BARKER AND JACK YARWOOD most obvious feature of the data is the almost total invariability of the spectral para- meters for a given salt. The band centre and width remain constant (within the experimental error) over a concentration range of 0.8-0.05 mol dm- and a temperature range of nearly 70 K for three different solvents. (There is a suspected tendency to show some increase in intensity as a function of increasing temperature but this effect if real is small.) On the other hand there is a considerable shift of vCAon changing the anion and this is roughly in proportion to the increase in effective reduced ma^^.^*^*^^ The data serve to confirm that this band is due to an internal mode (or modes) of the ion-pair or aggregate.They also enable us to confirm that the principal effect of the presence of these ions is as expected a large dipole-induced dipole effect on the surrounding solvent molecules. Within the framework of the stochastic model outlined above we might expect that this vibration would be sensitive to both temperature and solvent since we have treated the collisions of the solvent as a stochas- tic process-giving rise to k(t)of eqn (5)-and these processes are obviously dependent on temperature and microscopic solvent properties (the solvent properties are usu- characterised by a "friction constant " p obtained by using the Langevin ally20*21 equation of motion for the displacement coordinate rcA [i.e.fcA + picA+ &ArCA = F(t)]. Such a friction constant (or damping factor) will be temperature- and solvent-dependent). However it is easily shown that in the classical (high tempera- ture) approximation the root-mean-square amplitude of the oscillator A (equivalent to (k2)lI2 in the slow modulation limit and measured by the band width in this approximation) is proportional to T1/2so over a range of 70 K the effect is unlikely to be large. The damping factor p being a viscosity coefficient is exponentially dependent on temperature and should also change from one solvent to another.The fact that little or no change occurs in the band profile with temperature or solvent strongly supports the assertion that one is dealing with the slow modulation limit this is further rationalized as follows. It seems that bis so large34 for the concentra- tions studied so far that the vibrational mode vCAis heavily overdamped20*21 (p/2c0,A > 1). Since the amplitude of the modulation <k2)1/2 is also large we see that the slow modulation limit (A p/2co2,A % 1 in this notation)20*21 arises as a natural conse- quence of the combined high viscosity and rapid rate of vibrational relaxation.22 We note here that the mean vibrational lifetime measured by the rate of decay of q(t)-see fig. 6-has a time constant of -0.4 ps. This is comparable with the rate of collision of the solvent molecules of course and means that the rate of modulation of the environment of the ion-pair oscillator is a considerably slower process than the rate of solvent-solvent collisions.It is known from dielectric studies5p7 that the time between ion-ion collisions is of the order of 200 ps and it is possible that it is this quasi-static situation of the solute particles which leads to the slow modulation limit. What is clear is that the variation of k(t) [eqn (5)] is slow compared with the decay of ~(t) and that this is caused (as far as the model is concerned) principally by a very high viscosity coefficient p. Our current studies are aimed at collecting more accurate data at reduced concentrations and extending the solvent and temperature range so that some of these ideas may be further tested.As we have already pointed out the band " C " in the spectrum is explained within the framework of our model as a combination band of v + vcA which arises through strong coupling of the solvent collisional and ion-pair vibrational motions. Interestingly the band is very weak in (or absent from) the spectra of Bu4"N+Br- solutions (fig. 4) but one can only speculate at the present time as to why this is. Only an investigation of salts with a range of anions will indicate whether or not there is a strong anion dependence of the intensity of this band. Finally it is interesting to consider what aspects of the model described and the 146 STUDIES OF ION-ION AND ION-MOLECULE INTERACTIONS conclusions drawn here have a bearing on models used to interpret microwave data on similar solutions.Two points emerge which need to be carefully considered in relation to the data obtained at lower frequencies. These are (i) the very severe perturbation of the solvent molecules and their large induced dipole moments; (ii) the very large viscosity coefficient-arising very probably from the large macroscopic viscosity of the solutions. Both these effects are likely to be important in the micro- wave region of the spectrum and indeed have been at least implicitly taken into acco~nt.~*'-~~ The real value of this work may well be that we have shown a com- bination of studies in different frequency regions leads to a better understanding of the fundamental resonant and relaxation processes involved in the behaviour of similar or identical systems.The help of Dr G. N. Robertson (University of Cape Town) with the theoretical aspects of this paper is gratefully acknowledged. Thanks are also due to Beckman- R.I.I.C. Ltd. for their continued support (through the C.A.S.E. scheme) and to the S.R.C. for the C.A.S.E. award (to C. B.) and equipment grants. We are grateful to Dr. R. N. Jones for copies of the N.R.C. band fitting programmes. J. C. Evans and G. Y. S-Lo J. Phys. Chem. 1965,69 3223. 'M. J. French and J. L. Wood J. Chem. Phys. 1968,49,2358. 'C. Barker and J. Yanvood J.C.S. Faraday II 1975,71 1322. 'H. Cachet F. F. Hanna and J. Pouget J. Chim. phys. 1974,71,285 1546.J.-C. Lestrade J. P. Badiali and H. Cachet Dielectric and Related Processes ed. M. M. Davies (Specialist Periodical Reports Chemical Society London 1975) vol. 2 pp. 6-50. C. Barker Ph.D. thesis (University of Durham 1976). J. P. Badiali H. Cachet A. Cryot and J.-C. Lestrade J.C.S. Faraday II 1973,69 1339. J. P. Badiali H. Cachet A. Cryot and J.-C. Lestrade Molecular Motions in Liquids ed. J. Lascombe (Reidel Dordrecht 1974) p. 179. J. P. Badiali H. Cachet P. Canard A. Cryot and J.-C. Lestrade Compt. rend. C 1971,283 199. lo E. A. Cavell and M. Azam Sheikh J.C.S. Faraday IZ 1973,69 315. E. A. Cavell and P. C. Knight J.C.S. Faraday IZ 1972 68 765. l2 M. M. Davies and G. Williams Trans. Faraday SOC. 1960 56 1619. 13 J. Pitha and R. N. Jones Canad.J. Chem. 1966,44,3031; 1967,45,2347. l4 M. M. Davies Molecular Motions in Liquids ed. J. Lascombe (Reidel Dordrecht 1974) p. 615 and references contained therein. l5 P. L. James M.Sc. thesis (University of Durham 1976). l6 E. A. Moelwyn-Hughes,Physical Chemistry (Pergamon London 1961) pp. 303-4. l7 C. A. Kraus J. Phys. Chem. 1956,60,129. l8 C. J. F. Bottcher Theory of Electric Polarisation (Elsevier Amsterdam 1952) chap. 5. l9 Landolt-Bornstein Zahlenwerte und Funktionen (Springer Berlin 1951) Band 113. 2o R. Kubo in Fluctuation Relaxation and Resonance in Magnetic Systems ed. D. Ter Haar (Oliver and Boyd London 1962) pp. 23-68. 21 J. Yarwood and G. N. Robertson Mol. Phys. 1977 in press; Nature 1975,257,41. 22 S. Bratos J. Chem. Phys. 1975,63 3499; S.Bratos J. Rios and Y. Guissani J. Chem. Phys. 1970,52,439. 23 W. F. Edgell Ions and Ion Pairs in Organic Reactions ed. M. Szwarc (Wiley-Interscience New York 1972) vol. 1 chap. 4. 24 G. A. P. Wylie ref. (20) p. 7. 25 J. P. Badiali H. Cachet and J.4. Lestrade J. Chim. phys. 1967 64 1350. 26 M. J. Wootten Electrochemistry ed. G. J. Hills (Specialist Periodical Report The Chemical Society London 1973) chap. 2 and references therein. 27 G. W. F. Pardoe Trans. Faraday SOC. 1970,66,2699. 28 M. Davies G. W. F. Pardoe J. E. Chamberlain and H. A. Gebbie Trans. Faraday SOC., 1968 64,847. 29 M. Evans J.C.S. Faraday IZ 1975,71,2051 and references quoted therein. 30 G. J. Davies J. Chamberlain and M. Davies J.C.S. Faraday ZI 1973 69 1223. COLIN BARKER AND JACK YARWOOD 31 H.A. Stuart MoZekiiZstruktur (Springer Berlin 1967) p. 416-20. 32 S. R. Jain and S. Walker J. Phys. Chem. 1971,75,2942. j3P. Bacelon J. Corset and C. de Lo=,J. Chim. phys. 1973,70,1145. K. F. Denning and J. A. Plambeck Cnnad. J. Chem. 1972,50,1600. 35 J. A. Geddes and C. A. Kraus Trans. Faradzy SOC.,1936,32,585.

ISSN:0301-5696    

DOI:10.1039/FS9771100136

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. A. D. Buckingham (Cambridge University) said Deutch has drawn our attention to theoretical difficulties in obtaining reliable single-molecule orientation correlation functions from dielectric dispersion measurements on polar fluids. Similar problems have been encountered in the determination of dipole moments of molecules in condensed phases but these are significantly reduced by studying dilute solutions in a non-polar solvent. For this case E(W) differs from ~(0) by a small number proportional to the concentration of the polar solute and Deutch's eqn (2.10) and (2.11) become equivalent at infinite dilution. Does he believe that dilute solutions of polar compounds in non-polar solvents are suitable systems for measuring single-molecule orientation-correlation functions ? And could he be enticed into giving us some quantitative estimates of how serious the fundamental difficulties may be ? Prof.A. Gerschel (Orsay)said In response to Deutch I wish to make three points in favour of the dipolar absorption method. The first point deals with the choice between the Cole-Glarum or the Fatuzzo-Mason expressions. We remark that for computing statistical functions in the time domain representation only under certain conditions is there a significant departure between the results of both relationships which exceeds the experimental uncertainties. Such is the case with highly polar liquids possessing a large zero-frequency value co and intense absorption-dispersion features. In such liquids as recently noted with CH3F CH3Cl and CHF3' differences show up in the decay of correlation functions and memory functions depending on which of the theories was applied.I want to point out however that for the vast majority of liquids not possessing such very intense absorption features only minor differences appeared in the computed statistical functions (this was checked with liquids OCS and CHC13) still well within the experimental uncertainties. A second remark deals with the relationship between many-particle and single- particle correlation functions. One may agree that the search for a single-particle information is hopeless when one starts with the mixed information that is provided by dielectric relaxation. But since it is actually multimolecular information in- corporating both self and collective motion it is wise to keep it in this form and to complete the information from other " monomolecular," sources such as spectro-scopic studies of rotational broadening.Then one may gain by comparison information on the relative importance of collective motions. Taken from this point of view the useful and genuine character of dielectric relaxation is precisely to incor- porate both the effects of self and of cross correlations. Finally dielectric measurements may be useful in selecting models of molecular motion. Only if one restricts the measurements to dielectric relaxation are the pro- cesses insensitive to the details of the motion as a consequence of being essentially Markovian. But if dipolar absorption is considered as a whole it is seen to incor- porate the high frequency end that corresponds to damped librational motion.This latter gives information on short-time behaviour when the statistical functions are computed-correlation functions and memory functions-and it is a very sensitive test A. Gerschel C. Brot I. Dimicoli and A. Riou Mol. Phys. 1977 33 527. GENERAL DISCUSSION for any model to check its behaviour at any time against the actual time dependence extracted by Fourier inversion of the total dipolar absorption features. Typical examples of models not giving simultaneously acceptable agreement at short times and long times are the so-called "extended diffusion " models (after Gordon's J and A4 models).1*2 Models of the "itinerant oscillator " type3p4 give a much more satis- factory description on these grounds.Dr. G. Wyllie (University of Glasgow) said Although the qualifications set out by Deutch are important it may still be worth while to see whether there exist systems reasonably described by available theories. But one further qualification is that the dielectric friction term included in Deutch's eqn (3.3) is correct for small angle libration the case discussed by Fatuzzo and Mason wrong for free rotation even with a fixed axis (in which case successive rotations commute) and probably nearly correct for a diffusive motion with a small mean free angle of rotation. If then we combine Deutch's eqn (3.3) and (2.10) the function Eo -1 E2 + EoE + Eo 1 + icoz = -&o -(2E + l)(&-1) contains our information about the molecular mechanics with the dipole-dipole interaction more or less subtracted out.In the particular and probably rare situation where the dipole-dipole random force is uncorrelated with the residual random force acting on a molecule in small-angle diffusive rotation the related function should be the transformed dipole autocorrelation function for the (not realised) motion with the dielectric friction switched off and nothing else altered. Dr. W. Alexiewicz Dr. J. Buchert and Prof. S. Kielich (Poznari) (communicated) Had any of us been able to attend the Symposium we would have presented the fol- lowing material to supplement our paper read at this Symposium. The time-variations of the quadratic variations in electric permittivity in liquids take a simpler form if the analysing field is not time-variable or if the frequency of its vibrations is very low since now the dispersional factors R(uabc)behave as follows Eqn (17) now leads to the following simpler expressions A.Gerschel C. Brot I. Dimicoli and A. Riou,Mol. Phys. 1977,33 527. * J. O'Dell and B. J. Berne J. Chem. Phys. 1975,63,2376. N.E.Hill Proc. Phys. Soc. 1963,82,723. 'G.Wyilie J. Phys. C. 1971,4 564. GENERAL DISCUSSION which are especially well adapted to numerical analysis. In the approximation of Debye rotational diffusion one has r1 = 3r2,where 2, the relaxation time of electric birefringence,l can be used for the determination of the evolution in time of the various temperature-dependent contributions to dielectric saturation by a procedure similar to that applied with respect to the electric birefringence of liquid~.l-~ The curve shapes of the time-evolution of the temperature-dependent contributions reflect the competition between the purely dipolar term and the terms related with the aniso- tropy in electric polarisability of the molecule.This competition is the result of the differences in rise-time and sign of the various terms. The rise in time of the polarisa- tion due to reorientation of the permanent dipole moment is slower than the variations in time due to reorientation of polarisability anisotropy. In order to plot the time-evolution of dielectric saturation which sets in when ex- ternal electric fields are switched on to a system of non-interacting dipolar linearly polarisable molecules (m# 0 y # 0 b = 0) we introduce as in ref.(3) the following dimensionless molecular parameter r = (K) - (3) 1 m2 k3T3 k2T2 kT y ' Whereas for molecules with weak permanent dipoles (and non-dipolar ones) one has r !z 0 in the case of strongly dipolar molecules r differs considerably from zero. Obviously the sign of r depends on that of the anisotropy of electric polarisability of the molecule y = a, -axx. Numerical analyses of the time-evolution of a system of molecules carried out on the basis of eqn (19) are shown in fig. 1 (a) for some values FIG.1.-Time-dependence of the quadratic variations in electric permittivity in liquids calculated 1 m2 from eqn (19) for some values of the molecular parameter r = kT -Y .(a) r> 0,(b)r> 0. of r > 0 and in fig. 1 (b) for r < 0. One notes that competition between the molecular effects leads to large differences in shape between the curves. From eqn (19) the H. Benoit Ann. Phys. 1951,6,561. C. T O'Konski K. Yoshioka and W. H. Orttung J. Phys. Chem. 1959,63,1558. M. Matsumoto H. Watanabe and K Yoshioka J. Phys. Chem. 1970 74 2182; Kolloid-2 1972,250,298. M. Gregson G. P. Jones and M. Davies Trans. Faraday Soc. 1971 67 1630; B. L. Brown G. P. Jones and M. Davies J. Phys. D :AppI. Phys. 1974,7,1192. GENERAL DISCUSSION n saturation value attained by the system after a sufficiently long time (t z,) amounts to 4 Ae(0; t z,) -1 N m4 1_-_-(21) 45 Y (kT)3( Y Y’)* E2(0) Eqn (19) and (20) show moreover that for the two values of Y the steady state dielectric saturation vanishes.Fig. 1 (a,b) moreover shows that experimental determinations of the time-evolution leading to a steady state of dielectric saturation in a liquid can permit the clarification of its electrical nature. For a substance with known values of the molecular para- meters m and y an exact numerical analysis of the time-evolution of saturation can be made by having recourse to eqn (19). Other nonlinear effects can be analysed along similar lines. The latest experimental results achieved in the study of dielectric saturation in liquids1S2 stimulated us to concentrate on this effect as it now appears feasible to determine its evolution in time leading to the steady state.Dr. G. Williams (Aberystwyth) said In response to Wegdam’s comments the theory of the dynamic Kerr-effect was first formulated for the special case of small- angle rotational diff~sion,~~ when for the permanent dipole reorientation of an op- tically anisotropic molecule it was found that the rise-function for the birefringence is a weighted sum of decay-functions* exp-6Dt and exp-2Dt while the decay-function only involves exp-6Dt. Thus the decay-function is faster than the rise-function for this model of reorientation. In contrast a model of “fluctuation-relaxation ” yields5 M. Gregson G. P. Jones and M. Davies Trans. Furuday SOC.,1971 67 1630; B. L. Brown G. P. Jones and M. Davies J. Phys. D:Appf.Phys. 1974,7,1192. L. Hellemans and L. De Meyer J. Chem. Phys. 1975,63,3490. H. Benoit Ann. Physique 1951,6 561. ‘E. Fredericq and C. Houssier Efectric Dichroism and Electric Birefringence (Oxford U.P. London 1974). M. S. Beevers J. Crossley D. C. Garrington and 0.Williams J.C.S. Furaduy 11 1976 72 1482. D is the rotational diffusion coefficient. GENERAL DISCUSSION a rise and decay transient characterised by the same time-function which is not necessarily an exponential function of time. Thus a comparison of rise and decay transients for the Kerr-effect may give information on the mechanism for reorienta- tion. More generally the decay of birefringence may yield (Pz(cos 8 t)) while di- electric relaxation may yield (Pl(cos 6 t)).Comparison is made in the paper of these quantities for relaxation in the supercooled liquid state in order to elucidate the mechanism for reorientation of dipolar solute molecules. Although our data were obtained for fairly concentrated solutions so that internal field factors and cross- correlation terms may be important we do not think our present conclusions would be any different for more dilute solutions. Note that the permittivity and dielectric relaxation behaviour of fluorenonelo-terphenyl and tri-n-butyl ammonium picratelo- terphenyl solutions do not appear 1* to be made complicated by internal field and cross- correlation terms. We emphasize that In certain systems such factors are extremely important and must be taken into account in any comparison of dielectric and Kerr- effect data.Examples are (a)hydrogen-bonded liquids ;(6)flexible-chains in the liquid and solid states and (c) liquid-crystal phases. For (a)the equilibrium and dynamic aspects of cross-correlations between molecules is well known for the dielectric case3g4 but not for the Kerr-effect. For (b)the equilibrium aspects of dipole-correla- tions along a chain are well leading to an understanding of the static permittivity and Kerr-constant but the dynamical situation has received less atten- tion.8 For (c) the extensive angular correlations between molecules both in the liquid-crystal phase and above the clearing temperature lead to distinct differences (e.g. in relaxation time) between the dielectric and Kerr-effects (static and dynamic) which must be interpreted in terms of the long-range angular-correlations between molecules [see e.g.ref. (9) and ref. therein]. Dr. J. Yarwood (Uniuersity of Durham) said There are two possible effects on the measured absorption (and birefringence) which have been hardly mentioned (i) the effects of solvent polarisation caused by ion- or dipole-induced dipole interactions. Do such effects not contribute significantly to the dielectric absorption and if so is it possible to separate the different relaxation processes ? (ii) the effects of translational motions (i.e.,collisions) of the ion pairs or ion clus- ters. Models have been described by other authors lo which explicitly include linear Brownian motion between collisions. Can I ask why it is that such motion is not considered important for the systems considered here ? Dr.G. Williams (Aberystwyth)said The dielectric behaviour of solutions of tri- G. Williams and P. J. Hains Furdy Symp. Chem. SOC.,1972,6,14. * M. Davies P. J. Hains and G. Williams J.C.S. Faraday 11 1973,69 1785. N. E. Hill W. Vaughan A. H. Price and M Davies Dielectric Properties and Molecular Behaviour (Van Nostrand New York 1969). A. Rahman and F. H. Stillinger,J. Chem. Phys. 1971,55,3336. M. V. Volkenstein Configurational Statistics of Polymeric Chains-(Interscience New York 1963). P. J. Flory Statistical Mechanics of Chain Molecules (Interscience New York 1969). 'K. Nagai and T. Ishikawa J. Chem. Phys. 1965,43,4508. 'G. Williams and M. Cook Trans.Faraduy Soc. 1971,67,990. 9 M. S. Beevers and G. Williams J.C.S. Furuday 11 1976,72,2171. lo J.-C. Lestrade J. P. Badiali and H. Cachet Dielectric and Related Molecular Processes ed. M. M. Davies (Spec. Period. Rep. Chem. SOC. London 1975) vol. 2 pp. 106-150. GENERAL DISCUSSION 153 n-butyl ammonium picrate in non-polar solvents appears to be1-4 consistent with the majority of the solute being present as unassociated ion-pairs which undergo reorienta- tional motions. Thus for moderate concentrations (as studied in our paper) the dielec- tric relaxation for solvents of low and high viscosity may be considered to arise primarily from the vector dipole time-correlation function (p(0). p(t)> for the re- orientation of the ion pairs. Such an interpretation may not be applicable for the solute in solvents of high permittivity (polar solvents) where the solute ion-pairs may be partially dissociated into ions leading to dielectric relaxation involving ion-ion ion-dipole and dipole-dipole correlation function^.^ Clearly the mechanism for the reorientational motions of an ion-pair may involve an accompanying translational motion of the ion-pair.Put in another way the translational and reorientational motions are really coupled in a general equation of motion which cannot be separated into independent translational and rotational parts (e.g.,translational and rotational diffusion) since the elementary steps of both processes may occur on similar time-scales. For our present work the "fluctuation-relaxation" model does not exclude translational motion of a dipole when the dipole reorientates as a result of a local density fluctuation.Translational motions of dipoles with the exclusion of an accompanying rotational motion obviously cannot relax (~(0).~(t)). Thus in answer to (i) there may be a contribution from ion or dipole-induced dipole interactions but we would suggest that in our systems the major-part of the low frequency behaviour arises from (p(0). ~(t)) for the ion-pairs; and for (ii) translational motions are almost certainly involved in the reorientation process for ordinary dipoles and ion-pairs but the dielectric experiment yields the angular function (~(0). p(t)) = p2(cos 8 (2)). These comments relate to the presentation of dielectric data as ~"(w).For data presented as attenuation Nu) the features at higher frequencies than the dispersion of &(a)will be emphasised (since a(m) K &"/a).Thus information from a(co) will in practice relate to a shorter time-scale than information from &(a). a(m) yields (G(0) Xt)) while ~(m) where the former time-function yields (~(0).~(t)) Prof. J. S. Rowlinson (Oxford)asked Berne about the way in which the amplitudes of the long-time "tails " of the rotational velocity auto-correlation functions depend on the density of the fluid. Alder found that the tails on the corresponding transla- tional functions were most pronounced at densities intermediate between those of gas and liquid. S. M. Thompson G. Saville and I have made some molecular dynamic simulations of the gas-liquid surface of a Lennard-Jones fluid in which we find simi- larly enhanced amplitudes at long times for the translational motion of those molecules that start in the interfacial zone that is at an intermediate density and which move parallel to the surface.The densities in Berne's table 2.1 are also in this intermediate range and it would be of interest to know if this enhancement of the amplitude of the tail of the rotational function is also a characteristic of this range of density. Prof. 3.J. Berne (Columbia University) said In the rough sphere fluid as well as in fluids containing anisotropic molecules the correlation functions of the linear velocity M. Davies and G. Williams Trans.Faraday SOC. 1960,56 1619. E.A. S Cave11 and M. A. Sheikh J.C.S. Faraday II,1973,69,315. J.-C. Lestrade J P. Badiali and H. Cachet in Dielectric and Related Molecular Processes ed. M. Davies (Spec. Period. Rep. Chem. SOC. London 1979 vol. 2 p. 140. M. Davies P. J. Hains and G. Williams J.C.S. Faraday 11,1973,69 1785. ref. (3) p. 127. GENERAL DISCUSSION and angular velocity have the long time tails where d is the dimensionality and where av,dand are the amplitudes of the tails given by where n I D and v are respectively the number density moment of inertia Enskog translational diffusion coefficient and Enskog kinematic viscosity. These two ampli- tudes are related by aw,d = Ad(nav,d)2'd av.d where kid is a constant. For d = 3 Dorfman and Cohen have shown that n&,d as well as go through a maximum at intermediate densities.It follows that also goes through a maximum-albeit at a slightly different density than for av,d. In two dimensions the same reasoning is valid. It should be noted that the amplitudes above refer to time in units of the collision time to. This answers Rowlinson's question. Nevertheless I would like to take this oppor- tunity to draw certain distinctions between the behaviour of CV(t)and Cw(t). In the rough sphere fluid these two functions behave quite differently. At times short com- pared with the collision time both functions decay exponentially with time constants determined from the Enskog equation. For intermediate times that is for times longer than a mean collision time but shorter than the time at which the long time tails have set in C'(t) decays exponentially but more slowly than that given by the Enskog equation.This is what we mean when we say that Co(t)deviates positively from the Enskog correlation function. When In Cw(t)is plotted against the time in units of the mean collision time it is found that this positive deviation increases with density. This should be contrasted with the behaviour of CV(t). For intermediate times In Cv(t)always deviates negatively from the Enskog result. In the very dense fluid CV(t)displays a negative region. The negative deviation in CV(t)has been at- tributed to backscattering collisions. Recent mode-mode calculations (Desai et al.) show that these events do not effect CJt) and attribute the positive deviation in this function even at early times to the hydrodynamic coupling between spin and vortici ty .One way to compare the two functions is to compare the areas under CV(t)and Cw(t).These are DIDEand DR/DR,E respectively where D and DR are respectively the translational and "rotational diffusion " coefficients and DE and DR,E are the corresponding properties predicted by the Enskog theory. D/ DE increases with den- sity until it reaches a maximum at approximately a density equal to + the closest packed density. It thereafter diminishes to a value considerably below 1. This decrease springs directly from the backscattering events. At high densities the long time tails are too small to reverse this trend. DR/DR,E on the other hand should con- tinue to increase with density despite the fact that the asymptotic long time tail becomes progressively a smaller contribution.Prof. B. J. Berne (Columbia Uniuersity)said Deutch has raised a very interesting GENERAL DISCUSSION question concerning mode-mode coupling. There seems to be disagreement over the long-time tail in the orientational correlation functions Several calculations using mode-mode coupling schemes are in disagreement with each other. Noteworthy in this regard are the papers of Keyes and Oppenheim Garisto and Kapral and Pomeau among others. In all of these calculations the asymptotic behaviour is given by a power oft-’ which depends on the rank of the process. On the other hand the generalised hydrodynamic equations which successfully account for the splitting of the depolarised band in light scattering experiments can be used in conjunction with the asymptotic theory of Ernst et al.giving the result that for all (d+2) values of I these orientational functions behave asymptotically as t 2 . In addition if one uses a generalised rotational diffusion equation (Berne) one also arrives at this result. Deutch’s recent work with Hill applying the methods of Bedaux and Mazur raises the interesting possibility that this latter result arises from the hydro- dynamic field generated by a particle of finite volume whereas the mode-mode results have implicitly assumed velocity fields that are created by point particles. Clearly further work is required to resolve these differences.Dr. D. Frenkel (Amsterdam) said In the rotational relaxation of rough spheres angular momentum that is associated with the rotation of a given rough sphere may be transferred to two distinct (though coupled) angular momentum reservoirs in the fluid. Partly the angular momentum will be stored in the translational motions of the rough spheres (vortex currents) partly in the internal rotations. From both reservoirs angular momentum may leak back to the rough sphere under consideration thus slowing down its angular momentum decay rate with respect to the decay rate one would predict using Enskog’s theory. Could Berne please assess the relative import- ance of these two reservoirs for the slowing down of angular momentum relaxation? Prof. B. J. Berne (Columbia Uniuersity) said The total angular momentum of a fluid consists of two parts a part due to the translational motion of the molecules which gives rise to orbital angular momentum and a part due to the tumbling of the individual molecules which we call the spin angular momentum.The spin angular momentum of a molecule can relax only by exciting the orbital and spin angular momentum of the remaining degrees of freedom. This is required by angular momentum conservation. The orbital angular momentum created can be regarded as a vortex field. A relaxing particle therefore creates a vortex field which in a dense fluid diffuses away with a “ diffusion ” coefficient given by the kinematic viscosity. Some years ago Ailawadi and I showed that the long time tail is governed by this slow “ hydrodynamic ” diffusion of the vortex field and not by dissipation into the remain- ing spin field.This conclusion is strongly affirmed by recent mode-mode calculations. A striking example of this is that of a molecule relaxing in a solvent in which there are no spin degrees of freedom so that there can be no dissipation into the spin field. We have computed the correlation function for a non-uniformly rotating sphere with stick boundary conditions in a viscous continuum fluid. The long time tail in the angular velocity correlation function is identical to that calculated for a fluid with a spin field. Prof. Th. Dorfmiiller (Bielefeld)said The shear viscosity of liquid crystals as measured in a capilIary viscometer near the transition temperature exhibits a shape GENERAL DISCUSSION as shown in the figure.When the isotropic liquid is a few degrees above the transi- tion temperature it has a viscosity which is higher than the viscosity of the nematic liquid at lower temperatures. As the rotation of the molecules around the long axis is strongly hindered in the nematic phase as compared to the isotropic phase the viscosity does not seem to be simply a monotonic function of the single particle reorientational times. Thus we are faced with the situation that the single particle reorientation is faster in a high viscosity than in a low viscosity liquid. It seems that we have to accept that with a capillary viscometer we measure a collective reorientation time which under some circumstances (highly anisotropic molecules in an ordered state) does not behave as the single particle reorientation time.Photon correlation spectra of light scattered in many supercooled associated liquids show that in such systemsthe relaxation ought to be described in terms of a distribution of relaxation processes. Hence the question arises as to whether we can speak of a single relaxation time as in eqn (2) when dealing with the narrow central line at Our measurements on polarized and depolarized light scattering of supercooled polyalcohols always show a broad distribution of relaxation times conforming to a distribution function obtained by Lit0vitz.l According to Isakovich and Chaban2 we have to consider the light scattered from such systems as due to the fluctuation of a local order parameter.The time depend- ence of this is due to a simultaneous appearance of two coupled mechanisms a diffusional contribution to the correlation function P(r t) and a relaxational process both being governed by the equation P(r t) = DV2P(r t) -1 P(r t). TO Under certain realistic conditions this leads to a distribution function of relaxation times depending upon two parameters a characteristic relaxation time zoand a width parameter B. B has been shown to be extremely wide. The frequently used Cole/ Davidson distribution function less accurately describes the experimental results. Furthermore the temperature dependence of z is of the non-Arrhenius type and is practically identical to the temperature dependence of the macroscopic shear viscosity.C.J. Montrose and T.A.Litovitz J. Acuusf. SOC.Amer. 1970,47 1250. * M.A. Isakovich and I. A. Chaban Sou. Phys. JETP 1960,23,893. GENERAL DISCUSSION Dr. G. Searby (Nice)said We are of course quite aware that supercooled liquids show a distribution of relaxation times and Dorfmuller's query on the validity of the analysis used in our paper (and elsewhere) is quite legitimate. It is important how- ever to realise that in such liquids the width of the distribution decreases as the melting point is approached from below and can be quite narrow in the normal liquid state with a consequent nearly Arrehnius type of behaviour for z and q. Light scattering spectra in the HH geometry show a weak fine structure at the Brillouin frequency (arising from coupling between reorientation and longitudinal sound waves) but since this structure is displaced in frequency very localised and also very weak it may easily be ignored and the profile of the HH lines can be used to measure the average reorientation time and to estimate the distribution parameter p.We have analysed a number of HH spectra of different viscous liquids assuming a Cole-Davidson distribution of relaxation times. On the whole the results are as fol- lows (a)k2rl < 1 (generally above and away from the melting point). The distribution pr of relaxation times is too narrow to be evident over the limited frequency range covered by the interferometer. (b).%' > 1 (generally around the melting point).In some cases a distributed Pr spectrum provides a slightly better fit than a Lorentzian spectrum. For example for ethyl benzoate close to the melting point we found /I= 0.9. So the linewidths quoted at the lowest temperatures may indeed need to be interpreted with a little caution. The important point however is that in this region of 3the HH lineshape is nearly Pr Lorentzian wheras the VH lineshape is very different-see our fig. 7 for example. The small distribution of z therefore has no direct effect on the shape of the VH spectra. 2B 1 (supercooled liquids far below the melting point). In this region the Pr reorientation line is accessible to light beating techniques but is orders of magnitude narrower than the interferometer instrumental function.It is not possible to knalyse the spectra in terms of z q or R,nevertheless there re- mains an important difference between the observed VH spectra and the spectrum predicted by eqn (2). This difference is the integrated intensity of the propagating doublet which according to eqn (2) is equal to the integrated intensity of the reorienta- tion line independent of z q or R. It is probable that a more complete theory includ- ing the effect of one (or more) relaxing variables would account for the observed spectra. Prof. M. Davies (Aberystwyth)said Why did you heat your liquid crystal 140 "C above the clearing point before taking a spectrum? Dr. G. Searby (Nice) said The reason is very simple we wanted to verify that a liquid crystal in the isotropic phase shows the same coupling (between reorientation and the hydrodynamic modes) that is found in other liquids.The value of the coup- ling parameter in this case is also of great interest. We therefore chose the tempera- ture for which I? for the liquid crystal was around 1 GHz as for other liquids studied. k2r This would also give -a value of around 0.3 where the VH dip is most easily seen. Pr GENERAL DISCUSSION Because of the strong correlations that exist even in the isotropic phase the re- orientational motion is very slow and a very high temperature was necessary in order to obtain a motion sufficiently rapid. It also follows that at more moderate tempera- tures the ratio k2V -is very high as in supercooled liquids but in this case the vis- Pr cosity is very much lower and there should not be the complication of other slow variables affecting the spectrum.We would expect therefore that eqn (2) would accurately describe the VH lineshape for all values of % contrary to the case of Pr supercooled liquids. Dr. J. M. Vaughan (Maluern) said I would like to compliment the two previous speakers Searby and Pecora on their contributions which I have much enjoyed reading. While we are dealing with laser scattering studies it is perhaps appropriate to consider some developments in Fabry-Perot interferometry which will I think make a useful contribution to our subject over the next few years. In particular I am referring to the use of multi-pass instruments; these comments arise from collabora- tion with G.W. Bradberry of Exeter University. The instrumental form due to a single pass Fabry-Perot interferometer is given by the well-known Airy formula which is modified in practice by departures from plate parallelism flatness and the finite size of pinhole aperture etc. The extinction (ratio of instrumental peak height to mid order background) rarely exceeds lo3. With the technique of passing the light two or more times through the etalon the instrumental width near the line centre is made somewhat narrower but most importantly the inter-order background is greatly reduced. Extinctions greater than lo6are routinely obtained in for instance triple-pass use. So much is well known and the obvious applications have been made to the study of very weak spectrally-shifted features in the presence of for example strong elastic scattering.This is illustrated by fig. 1 and 2 which are taken from some recent investigations of ours on Brillouin scattering in bulk and thin film samples of cyanobiphenyl liquid crysta1s.l. -1.5 GHz ln 4 3 V I 00 frequency FIG.1.-Quasi-elastic light scattering spectrum of a thin film sample of 5 CB at 32.5 "Cin the ne- matic phase; the interferometer recording shows channel contents (photon counts) plotted against frequency. The Rayleigh peak has been reduced on the diagram by a factor of lo5. The Brillouin peaks are shifted rt4.76 GHz and the scattering vector is of magnitude 1.82 x lo5cm-' However the point of my comment is not so much concerned with these Brillouin scattering studies it is rather to draw attention to the possibilities opened for the study of Rayleigh lines by exploiting the high extinction properties of the multipass instru- ment.With suitable choice of experimental parameters we believe that very sensitive analysis of the line shape is possible. In particular for lines of a Lorentzian charac- G. W. Bradberry and J. M. Vaughan. J. Phys. C:Solid State Phys. 1976,9,3905. * J. M. Vaughan Phys. Letters 1976 58A,325. GENERAL DISCUSSION temperature / 'C FIG.2.Brillouin shift hypersonic speed and attenuation in the region of the nematic-isotropic transition (dotted line) for 5 CB at a constant scattering vector of magnitude 1.82 x lo5 cm-'. ter or having appreciable wing intensity while the profile near the line centre may be extensively modified by the instrumental form because of the very high extinction the shape away from the centre is very little changed over a wide range of frequency.If a simple Lorentzian form can reasonably be assumed then an experimental measure- ment of its width r (half width half height) may readily be determined from the expression k2 r=nA.-T.Av where in an experimental spectrum A is the signal content within a small frequency interval Av at a distance kfrom the line centre and Tis the integrated signal within the whole spectrum. The fit to a single Lorentzian can obviously be examined by com- paring values of r obtained at different values of kin the same spectrum.By varying temperature / 'C FIO.3.Lorentzian line width r of the Rayleigh peak plotted against temperature for 5 CB in the isotropic phase. The clearing temperature of the sample is indicated by the arrow. The error bars are approximateIy f3%. GENERAL DISCUSSION the etalon plate separation the shape may be examined over an even larger range of frequency. The precision that may be realised is very high-for example the con- ventionally defined resolving limit of spectroscopic instruments is rarely smaller than 10 MHz. Problems of deconvolution near the line centre ensure that intrinsic Ray- leigh line widths less than this can be obtained only with very limited accuracy. However by applying the technique outlined to analysis of the wing intensity these limitations are overcome.Fig. 3 shows data obtained on depolarised Rayleigh spectra of 4’-n-pentyl-4-cyanobiphenyl(5 CB) in the isotropic phase close to the isotropic- nematic transition. Each value of r has been obtained from the intensity at twelve points in the frequency range 0.45-0.9 GHz. Within experimental error these values were the same showing a good fit to a single Lorentzian over this range; the statistical error in l? amounts to about &3%. The conventionally defined resolving limit (full width at half height) for the spectrometer was -100 MHz nearly two orders of mag- nitude greater than the smallest line width measurement. The technique is described in greater detail in a paper of Bradberry and myself.’ In conclusion there seems a good prospect that the multipass interferometer will be as fruitful in the precision analysis of Rayleigh lines as it has been in the study of Bril- louin scattering.Prof. B.J. Berne (Columbia University) said To take up a point raised by Manse1 Davies it now appears that hydrodynamic models can be applied with great success to the motion of molecules in solvents where the solvent molecules are not large com- pared to the molecule in question. The evidence for this springs from several sources. Zwanzig and Bixon have computed the velocity correlation function of a sphere moving in a compressible viscoelastic fluid. This agrees well with molecular dynamics studies of the smooth hard sphere liquid and of the Lennard-Jones liquid when slip boundary conditions are used.Perrin calculated the translational and rotational diffusion coefficients for ellip- soids with stick boundary conditions. When applied to the tumbling of small mole- cules like benzene the rotational diffusion coefficients disagreed with experiment by as much as one and two orders of magnitude. Recently Zwanzig and Hu using a variational principle have calculated these quantities for ellipsoids with slip boundary conditions. Their results are in close agreement with the depolarised light scattering measurements of Pecora et al. on several small molecules. Acrivos et al. have im- proved on the agreement by introducing a more detailed model of benzene in which six slippery spheres are placed at the vertices of a hexagon.Thus it appears that for small molecules hydrodynamics with slip boundary conditions is quite successful. In hydrodynamic calculations the fluid is treated as a continuum. The important physical assumptions are connected with the boundary conditions. One of the im- portant remaining problems is to show how the boundary conditions spring from the intermolecular forces. This is actually a subtle problem. The rough sphere fluid is a good case to consider. We find that although the spheres are microscopically rough stick boundary conditions are not applicable. It appears that the velocity field produced by a rotating sphere outside of its boundary layer is much more like that of a sphere with slip boundary conditions that that of a sphere with stick boundary conditions.Much remains to be done before we under- stand the underlying molecular mechanism. G.W.Bradberry and J. M.Vaughn Opt. Comm. 1977,20,307. GENERAL DISCUSSION Prof. R. Pecora (Star3ford Uruirersity) said In response to remarks by Mansel Davies The viscosity used in our studies is the macroscopic solution viscosity as measured by an Ostwald viscometer. There is in fact a great deal of evidence that hydrodynamic type theories (" general-ised hydrodynamics ") are applicable to phenomena at the molecular level. One strik- ing example of this is the molecular dynamics calculation of Alder et aZ.l These authors calculated both the translational self-diffusion coefficient and the viscosity of a hard sphere liquid and showed that the Stokes-Einstein relation between these quanti- ties calculated with dip boundary conditions was obeyed.We have performed measurements of rotational relaxation times of molecules in the liquid state as a function of solution viscosity. For classes of these liquids which satisfy the conditions (1) the solute and solvent molecules are roughly the same size (2) there are no strong intermolecular interactions (e.g. dimer formation hydrogen bonding) it is found that the single-molecule reorientation time has a vis- cosity dependence whose slope can be predicted by hydrodynamics with slip boundary conditions to within about &15%. The Stokes-Einstein expression with stick boundary conditions predicts much too much friction and is as you point out useless for predicting rotational relaxation times in this class of liquids.The Stokes-Einstein relation with stick boundary con- ditions is however valid for the rotational relaxation times of solute molecules which are much larger than the solvent molecules. The example you give of some extra polymer dissolved in a solution of rotating smaller molecules does not fit into either of the above categories. I would certainly not expect the rotational relaxation time to follow the macroscopic solution viscosity in this case. Rotational relaxation times of molecules near a gas-liquid critical point or a solution consolute point would probably not have simple relations to the macro- scopic solution viscosity either. Prof. D. Kivelson (Uniuersity of California) said In answer to the question of Mansel Davies concerning the use of solvent solution or microviscosities in making use of the Debye expression I would like to refer to and expand upon the discussion below eqn (25) in my introductory article.I for one shy away from the use of microviscosity (vmicro)because it cannot be measured independently of the Debye relationship-it is essentially a quantity with the dimensions of viscosity which ensures that the expression works exactly. As I have indicated I prefer to substitute tpc for qmicra where q is the independently measured macroscopic viscosity and K is the dimensionless coupling of rotational to translational modes discussed near eqn (23)-(25) of my introductory article. For small molecules that are not too asymmetric or interactive such as the xylenes discussed by Pecora K is small independent of Tand q and has values close to the hydrodynamic slip values.In dilute solutions of larger more interactive probe molecules K is still relatively independent of T,q and pressure but has values between the slip and stick values values that vary with solvent. If the solvent consists of two components for many solvents the macroscopicsolution viscosi$y can still be used with the Debye expression provided an appropriately weighted K,such as ([dA)-IC(~)]X~ + B. J. Alder D. M. Gass and T. E. Wainwright J. Chem. Phys. 1970,53 3813. * J. R. Bauer J. I. Brauman and R. Pecora J. Amr. Chem. SOC.,1974,96,6840. GENERAL DISCUSSION dB)) are the K'S for the probe molecule in pure A and pure is used where dA)and dB) B respectively and X is the mole fraction of the A constituent in the solvent.Once again this effective K is independent of T and q. Davies has pointed out that if small amounts of polymer are added to certain solu- tions the solution viscosity changes by orders of magnitude but the rotational correla- tion time q is relatively unaffected. Of course as he maintains the Debye expression is not useful in this case since the solution viscosity appears to have little connection to zi. As indicated in my introductory article I would propose the following explana- tion of the various situations mentioned. Coefficients of shear viscosity are macro- scopic in the sense that they depend upon the interactions and structure over a distance of many molecular radii; however computer calculations seem to indicate that this distance may not be much greater than ten or so radii.Similarly the rotational relaxa- tion of a molecule depends upon more than its interactions with its nearest neigh- bours since in turn the nearest neighbours are severely constrained by their neigh- bours. It is not unreasonable to assume that for rotational relaxation the structure and interactions over a distance of many molecules a distance comparable to that characteristic of viscosity determines the actual behaviour. However if the viscosity is enhanced by the addition of polymer one would find that the viscosity increases with wavelength up to wavelengths of several hundred Angstroms while the rotations of small molecules in fluid cavities are still relatively independent of wavelength for wavelength above about 20 A.This dependence upon wavelength was emphasized by Deutch in his comments. The coefficient of shear viscosity is normally measured at very large wavelengths. In summary we can assume rl(Ar) and q(A,) where Ax and Av are wavelengths above which zi and q respectively are independent of wavelength. The Debye relation states that zi(A,) is proportional to q(A,); if A,, 5 AT the solution viscosity ~(oo) may be used but if Aq A, a microviscosity q(&) must be used. In " normal " situations A,, E & but with the addition of polymer it is likely that A is very large. Dr. P. S. Y. Cheung (University of Kent) said I would like to suggest an alternative method to interpret Pecora's results for the neat liquids.The comment is however directed to Kivelson since it concerns the general usefulness of eqn (2) in the text Eqn (2) is derived from the Mori projection operator method and applies when the orientational variables are " slow ".I When applying eqn (2) two questions arise (1) " When are the relevant variables ' slow ' enough?" and (2) " How does one interpret the dynamic factor gN,which is formally defined in terms of a projection operator in terms of microscopic processes?'' (In other words what information can be gained from a knowledge of gN?). By recasting eqn (1) into a slightly different form it is possible to interpret the z,/z, data outside the projection operator scheme.We define the normalised correlation function for light scattering as and those for the single-particle and distinct contributions as T.Keyes and D. Kivelson J. Chem. Phys. 1972,56,1057. GENERAL DISCUSSION wherefN = (N -1)(~(’)(0)~(2)(0)) is the same as theflv in (2). Now if we define the correlation time z of a normalised correlation qx(t)as then where h is the ratio zd/zSpand contains information relating to correlated motion. Thus eqn (B) allows one to interpret the experimental data for zLsand zspin terms of two well-defined microscopic parameters fN and h. The equation is quite general and does not depend on any model or theory of reorientation. A corresponding equation applies for dielectric relaxation and the present view of experimental data is close to comments made by Williams in his paper presented earlier in the meeting.Using Pecora’s results for zLs zspand fN for the neat liquids one finds 0-P-m-xylene. h 15 & 65 0.4 & 0.2 3.0& 1.5. On general grounds one does not expect Td to differ from zspby a factor larger than 10 or less than 0.1 so the value for o-xylene is probably too large-the large un- certainty arises because f N as quoted carries a large experimental error. The interpretation of h is by no means straightforward and is too complex a problem to enter into in detail here. One approach would be to evaluate typical values for specific types of correlated motion or by computer simulation for general molecu- lar types classified according to shape or the interaction potential so as to facilitate interpretation of experiment.Incidentally a computer simulation of a nitrogen like diatomic by Levesque and Weisl gives fN = -0.09 and h -4. It is interesting to note that for the nitrogen simulation eqn (2) is invalid because the correlation functions are non-exponential with a corresponding non-Lorentzian Rayleigh spectrum. Indeed if one goes far enough out into the “ wings ” the light scattering spectrum of benzene is non-Lorentzian showing significant free-rotation even at room-temperature.2 Thus for xylene one should be careful to include the wings to obtain zLs as defined by eqn (A)-this necessitates of course considerations of induced ~cattering.~ We note that Pecora’s zLsis deduced from the Lorentzian part of the spectrum which is correct only within the framework of eqn (2).In view of the above comments the usefulness of eqn (2) in analysing experimental results is questionable. My view is that the most important result of Kivelson’s paper is not eqn (2) but lies in the explanation it gives for the occurrence of a Lorent- zian in the centre of the Rayleigh spectrum even in the presence of correlation of orientation and correlated motion. D. Levesque and J. J. Weis Phys. Rev. A. 1975 12,2584. * H. D. Dardy V. Volterra and T. A. Litovitz J. Chem. Phys. 1973,59,4491. P. E. Schoen P. S. Y. Cheung D. A. Jackson and J. G. Powles Mul. Phys. 1975,29 1197. GENERAL DISCUSSION Prof. D. Kivelson (Uniuersityof California)said In response to Cheung's ques- tions we can of course write the many particle rotational correlation time zl as 21 =zpq1 +X] where z~(~) is everything else that enters is the single particle correlation time and XZ~(~) into zZ.However this expression merely states that zl and zl(s)differ and it does not relate the difference to quantities that can be measured by other methods. The statement becomes particularly useful if Ng z 0 because 1 +Nf can be determined inde- pendently; furthermore the dynamic quantity g in the diffusion limit is related to a physically significant rapidly decaying correlation function. Prof. B. J. Berne (ColumbiaUniversity)said The problem discussed by Keyes and Kivelson is the following Given that CI@)(t), the self-orientational correlation func- tions decay exponentially with time constants z~(~), how do the collective correlation functions Cl")(t) decay.Their solution to this problem is now well known to be The collective correlation function is also an exponential with a time constant related to the time constant appearing in the self motion Here g is a dynamic contribution arising from the cross correlation function of the angular velocities (weighted by orientational terms) andfis a static orientational struc- ture factor. It is tempting to subdivide CltN)(t)into two terms with different time constants one arising from the self motion and one arising from the distinct motion. In my opinion this would lead to certain difficulties. The self term is always positive whereas depending on molecular shape the distinct term might be negative for certain ranks 1.This leads to the possibility of negative power spectra. Aside from this we should take our cue from the vast literature in hydro- dynamic and generalised hydrodynamic fluctuation theory which deals with the trans- lational analogues Fs(k,t) and F(k t). The pole structures of these two functions are quite different. Only at very large values of k does F(k t) behave like F,(k,t). Moreoever at small and intermediate values of k the poles in the Laplace transforms of these functions are totally unrelated. At small k,Fs(k,t) is a single exponential and F(k t) consists of three exponentials. The damping rates in F(k,t) are unrelated to those in F,(k,t). Decomposition of F(k,t) into a self and distinct term would be fruitless.Consider the simple case of N solute molecules moving in a solvent. Then let GENERAL DISCUSSION @>,(k,t) and @(k t) be the translational self and collective correlation functions. Then using irreversible thermodynamics one can show that at small k OS(k,t) decays exponentially with time constant l/z = k2Dswhere D is the self diffusion coefficient whereas @(k,t) decays exponentially with time constant l/zN = k2L($) where L T’P is a kinetic coefficient and ap/acis a thermodynamic derivative of the chemcal poten- tial with concentration. At sufficient dilutionL 2i 0,but in general L = (1 + Ng)D where Ng represents a dynamical correction to the self diffusion coefficient.Also 5= lim S(k) afl k+O where S(k)is structure factor of the solute. It is the analogue of 1 + Nfin the orienta- tional problem. Thus there is a one to one correspondence here to the orientational problem. Dr. J. M. Vaughan (MaZvern)said I agree with Kivelson that the strong scattering close to the isotropic-nematic transition makes the spectroscopic task easier; and in the present work we have been able to employ low laser power to avoid heating the sample. However we have used the multipass technique quite successfully well away from the transition where the scattering is greatly reduced (and incidentally we have also observed the central dip in the V-H spectrum of 5 CB referred to by Searby). With a good stable interferometer and suitable choice of instrumental parameters such as laser power etalon aperture etc.I would say the technique can be employed with a wide range of scattering systems. Dr. J. Yarwood (University of Durham) said In connection with the dependence of vibrational band width on the upper state quantum number n,we have recently meas- ured the ratio of isotropic Raman band widths of fundamental and first overtone for three simple (polar) liquids and their dilute solutions in carbon tetrachloride. The results shown in the table,l p2 clearly indicate that this band width ratio has neither n nor n2 dependence. Furthermore it is clear that different modes have different ratios and also show different degrees of narrowing on going to a dilute solution. Thus for polar molecules it appears that the:situation may be more complex and this is of course not unexpected since in both Madden’s treatment and in that of Bratos et aL3 the effects of induced moments are neglected.It should also be noted that for acetonitrile and for nitromethane the effects of resonance energy transfer (i.e. coupling of transition moments of different molecules in the liquid) as shown4 by isotropic dilution experiments are very small. I would also like to present the results of some neutron quasi-elastic scattering data which we have recently made for acetonitrile and its solution in carbon tetrachloride. These results (obtained in collaboration with T. C.Waddington and P. G. Woodcock) were collected at low momentum transfer (Q I I A-l) and high energy resolution (180 peV) on the 4H5 spectrometer at A.E.R.E.Harwell. A plot of the quasi- elastic peak half-width (after removal of the ‘‘instrument ’’ width) against Q2 at A more complete table is given in J. Yarwood and R. Amdt Chem.Phys. Letters 1976,45 155. G. Doge 2. Naturforsch. 1973,28A 919; R. Amdt G. Doge and A. Khuen Chem. Phys. in press. S. Bratos and E. Marechal Phys. Reu. 1971 A4,1078. J. Yarwood R. Arndt and G. Doge Chem. Phys. (submitted for publication) and unpub-lished data. GENERAL DISCUSSION TABLE SHOWING COMPARISON OF FUNDAMENTALAND OVERTONE BAND WIDTHS FOR SOME SIMPLE LIQUIDS. (293 K). 1.41 1.30 2.48 1.61 CHJ v,/~v 2.88 3 .OO ~~ CH3N02 ~212~2 2.40 2.38 v3/21.'3 v4/2v4V5/21.'5 1.89 2.63 2.82 1.70 2.33 3.17 Concentration is 0.25 mole fraction of solute.low Q values gives a value of the translational diffusion coefficient D according to AE+(Q. E) =2hD;Q2. The data when plotted this way (see diagram below) show that D decreases drastic- ally on diluting with carbon tetrachloride (the data give 0;= 3.1 x cm2s-l for -_-__ FIGuRE.-comparison of Quasi-elastic peak width against Q2plots for CH3CN CH3CN in CD3CN and a 40% (by volume) solution in carbon tetrachloride. (Resolutionis about 180-200 peV at 1220 ps rn-l incident time of flight). GENERAL DISCUSSION the liquid at 295 K and D = 1.9 x low5cm2s-' for a 40% solution in carbon tetra- chloride). These data show that the collision rates increase on dilution with CC14 giving a slower rate of translational motion.This is entirely in accord with recent results we have obtained from the isotropic Raman scattering of the v band in the same systems. Here the values of the correlation time 2 obtained by fitting the ob- served vibrational correlation functions to the expression Y"(t) = exp -W">CtG + 2 (exp (-m-113) show a considerable decrease on diluting with CC14 (from 0.4to 0.25 ps). Such cor- relation times support the idea of an increase in the rate of hard collisions producing an increase in the rate of vibrational relaxation. It remains to be shown whether or not such changes in the dynamic properties of the acetonitrile molecules may be correlated with macroscopic viscosity changes or whether microscopic changes (due to changes in intermolecular potential) dominate the picture.Dr. P. A. Madden (Cambridge Uniuersity) said One of the reasons for developing the resonance Raman technique for vibrational dephasing studies was that spectra of very dilute solutions of non-polar diatomics can be obtained. For such systems the term in the Hamiltonian responsible for dephasing should be of the form Gq2(see the paper) the consequences of this term for the bandwidths can be evaluated in the slow and fast motion limits. For concentrated solutions of diatomics additional terms of the form Gi,Jqiqj(i # j) arise in which i andj label molecules for concentrated solu- tions of polyatomics the general term will be Gal:,pjqarqpj (i # j i =j) in which qai is the normal coordinate for the uthnormal mode of the ithmolecule.A recent study' (see also the comment by Lynden-Bell at this Symposium) shows that the effects are often non-additive. Complex interference effects may occur. It may be expected that when several of these terms are important the simple n and n2relationships will not hold. Similar relationships to those reported here by Yarwood have been found for other polyatomic systems.2 Dr. R. M. Lynden-Bell (Cambridge University) said I should like to amplify the refer- ence to work on vibrational dephasing3* that Madden made.5 In a neat liquid in the rapid motion limit where fluctuations in the intermolecular potential are fast com- pared with the vibrational dephasing time there are two types of term which contribute to dephasing. These are self terms such as and for an anharmonic oscillator and exchange terms where the vibrational quantum is exchanged between identical molecules 1 and 2.R. M. Lynden-Bell Mol. Phys. to be published. C. Brodbeck I. Rossi Nguyen-Van-"hanh and A. Ruoff Mol. Phys. 1976,32,71. P. A.Madden and R. M. Lynden-Bell Chem. Phys. Letters 1974,38,163. R. M.Lynden-Bell Mol. Phys. 1977,in press. P.A. Madden this Symposium. 168 GBNERA L-D I S CUSS 10N In these expressions V12is the intermolecular potential of molecules 1 and 2 and qj q2are the vibrational normal coordinates of these molecules. Vibrational dephasing is observed in isotropic Raman scattering (an I = 0 process) and in infrared (I = 1) and depolarized Raman scattering (I = 2).These dephasing times may differ as cross terms between exchange and self terms can con-tribute to I = 0 but not to I = I and negligibly to I = 2 giving T2-I = (Fez+ F2)r; I = 1 or 2 2y1= (F + F$7; I = 0 where Fe and F,are the amplitudes of the frequency changes due to exchange and self terms and z is the correlation time. Evidence for the importance of exchange effects in some normal modes comes from isotopic dilution studies. There are many different terms in the intermolecular potential which may cause vibrational dephasing short range forces dipole-dipole van der Waals etc. which have different angular and distance dependences. These contribute independently to the rate of dephasing and have different correlation times z. In solutions pair interaction of molecules of different types gives self terms but the exchange terms are no longer resonant.In a mixture the vibrational dephasing rate has the form where 7AA and xAB are correlation times for AA and AB intermolecular inter- actions. Tabisz and I1have tested some experimental results to see if they agree with these rapid motion predictions. First the measurements of methyl iodide vibrational line widths (v3(alg))by Campbell Fisher and Jonas2 asl(a function of pressure. Assuming that changes in z are proportional to changes in bulk viscosity and that changes in F2are proportional to changes in density one predicts that T2-l should be proportional to qp. We obtain from the experimental results good straight lines with a non-zero intercept i.e.T2-l = ko + Sqp where the slope S increases with temperature and the intercept ko,(intrinsic line width?) is apparently independent of temperature and is about half the total relaxa-tion rate. Data of Neuman and Tabisz3 on the vibrational dephasing of alg in benzene (992 cm") in liquid mixtures with CC14fit a relationship T2-I = ko 4-A ~AIDAA 4-A$ ~BIDAB where we have used the inverses of the self and mutual diffusion coefficients DAA DAB as measurements of the correlation times and assumed that F2AA and FA,are proportional to the molefractions of A and B MAand MB,respectively. Mr. M. R. Battaglia Mr. T. I. Cox Dr. P. A. Madden and Mr. R. A. Shatwell (Cambridge University) said The experimental results on vibrational dephasing quoted in the paper were obtained by workers not primarily interested in relaxation studies.In order to demonstrate that meaningful relaxation parameters can be ex- R.M. Lynden-Bell and G. C Tabisz 'Ckem.Phys. Letters 1977 in press, 'J. H. Campbell J. F. Fisher and J. Jonas J. Chem.Pks. 1974,61,346. 'M. N. Neuman and G. C. Tabisz Chem. Phys. 1976,15,195. GENERAL DISCUSSION 169 tracted from Resonance Raman lineshapes it is necessary to solve the problems of (a) providing a well defined baseline for the spectrum; (b) defining the medium temperature at the laser focus in an absorbing medium; (c) separating the hot band contribution from the band profiles. To this end we will describe our recent studies on the system I2 + CCl previously studied by Kiefer and Bernstehl The difference system described in the paper has been modified2 to deal with the problem described there of excess solvent scattering from the pure solvent half-cell.The light now passes through a Pockels cell followed by a polarizer (in the plane of the initial laser polarisation). A voltage is applied to the Pockels cell only when the pure solvent half-cell is illuminated; in such a way as to reduce the laser intensity incident on the cell. By varying the applied voltage the solvent bands may be nulled in the difference spectrum and a pure solute Raman spectrum obtained distorted only by (weak broad) fluorescence. The medium temperature was obtained from the ratio of the Stokes to anti-Stokes intensities of the CCl bands of the solution (adjusted for spectrometer response) and assuming efficient V -T relaxation.It was found to be 320 K (consistently for all CC14 bands) when 500 mW of 514.5 nm radiation was focus- sed on the mol dm-3 I2 + CCl solution. The Stokes to anti-Stokes intensity ratio for the I2 fundamental however showed a “ temperature ” of 284 K. The first (-21 1 cm’l) anti-Stokes line is the -1st member of the hot-band overtone pro-gression (Zl +n) and this result shows that the iodine hot-band intensities through- out the spectrum will not take their thermal values. Consequently when fitting a line to obtain the lineshape parameters the hot-band intensities (and their widths) must be allowed to vary freely. Shown in the figure are the square roots of the line widths of the 0 -+ n transi-n FIGuRE.-The square root of the width A,,/cm-I of the 0 3n line against n.The dotted line passes through the most reliable data points. tions obtained by fitting Voigt functions to the overtone profiles plotted against n. Separate Voigt functions were used for the fundamental (Io+n) and hot-band (Zl+n+l) contributions to a band with freely varying position width and height. The spectra were of sufficiently good quality that the positions of the hot-bands as W. Kiefer and H. J. Bernstein 3. Raman Spectr. 1973,1,417. * M. Battaglia T. I. Cox,P. A. Madden and R. A Shatwell in preparation. 170 GENERAL DISCUSSION fitted agreed closely with the values predicted by differencing the appropriate funda- mental positions and the widths agreed with the theoretical relationship to the funda- mental widths,l at least for the lower overtones (n < 5).For small to intermediate n values the widths closely follow the n2dependence discussed in the paper appropriate to the "fast motion " limit. But as the linewidth becomes large (n > 6) the n de-pendence is slower than n2 and is tending to the linear n dependence appropriate to the slow motion limit.2*3 Prof. A. Gerschel (Orsay)said A first remark is made to support the contribution of two-body encounters effects in liquids although we do not pretend to such a com- plete analysis as Litovitz's. We have considered the integrated intensity variations of far infrared absorption in compounds like CSZ4and CC14 that were studied in a range of densities and temperatures extending from the triple point up to the critical point.Analysis of the density dependence of these intensities was carried out and revealed acceptable agreement with a squared densities law. Since in the dense structure the frequency of collisional events also depends on the squared density we thought it reasonable to relate the variation of absorption intensities to that variation in the occurrence of collisions. In this respect a restriction of consideration of two- body collisions appeared satisfactory within the experimental accuracy. At longer times a second remark applies dealing with evidence of many-body effects in the molecular interactions at liquid densities as clearly revealed by far infra- red experiments on polar liquids especially with molecules possessing high permanent dipole moments.It comes as an effect of the local structure to suppress the randomiz- ing character of gas-like bimolecular encounters rather we should speak of corre- lating collisions to account for the persistence of a damped oscillatory librational m~tion.~*~ Narrower spectra are observed a reason for this being that advocated by Litovitz that many-body correlations decay slower than two or three-body correlations; the main reason however is here a consequence of increased coherence in the angular motion. Not surprisingly the measured effects of the many-body correlations were weak in the liquids investigated in this work constituted of atoms or spherical molecules since the advent of these effects depends on the degree of local organization a consequence itself of molecular anisotropy (of either steric or dipolar origin).Dr. G. C. Tabisz and Prof. A. D. Buckingham (Cambridge University) said An aim of the paper of Stuckart Montrose and Litovitz is to compare the collision- induced scattering from inert-gas atoms with that from tetrahedral molecules. We suggest a mechanism which can account for two distinctive features of the scattering from tetrahedral molecules namely the excess intensity of the depolarized scattering over the predictions of the (point) dipole-induced-dipole model and the occurrence of an extremely broad high-frequency tail.'. * This mechanism yields a collision-induced rotational Raman scattering.P. A. Madden and R. M. Lynden-Bell Chem. Phys. Letters 1976,38,163. S. Bratos and E. Marechal Phys. Rev. A 1971,4,1078. G. Doge 2.Naturforsch. 1973 28A 919. I. Darmon A. Gerschel and C. Brot Chem. Phys. Letters 1971,8,454. A. Gerschel I. Darmon and C. Brot Mol. Phys. 1972,23,317. A. Gerschel C. Brot I. Dimicoli and A. Riou MoZ. Phys. 1977,33 527. D. P. Shelton and G. C. Tabisz Proc. Fifth Int. Con$ on Raman Spectroscopy ed. E. D. Schmid (Freiburg 1976) p. 382. * F. Barocchi and M. Zoppi Proc. Fifth Int. Conf. on Raman Spectroscopy ed. E. D. Schmid (Freiburg 1976) p. 386. GENERAL DISCUSSION 171 Tetrahedral molecules lack a centre of symmetry and therefore possess a non-zero anisotropic dipole-quadrupole polarizability tensor A.l To appreciate the signifi- cance of A consider a light beam incident upon a pair of molecules 1 and 2 separated by a distance R.The electric field E associated with the light beam induces a dipole moment p = alE in 1 and the field F due to pl induces a dipole moment aaF in 2. This yields the well-known dipole-induced dipole (DID) contribution in R‘3 to the polarizability of the pair. The field-gradient F‘ due to pl also acts on 2 to induce a dipole moment +Az:F‘; moreoever the external field E induces a quadrupole moment 0 = Al . E in 1 and its field induces a dipole in 2. These two interactions yield a pair-polarizability varying as R‘4 and rotating with the tetrahedral molecules. The following expressions are obtained for the mean-square values of the pair-polariza- bility components a, and ayz,averaged isotropically over all orientations The first term in {a&> gives the polarized Rayleigh scattering; the terms in are the dipole-induced-dipole contributions while those in R-are new and yield the induced rotational scattering.For tetrahedral molecules A is determined by a single parameter A.l For CH, A(4n~J-lhas been estimated by Isnard et aL2 to be 2.35 x cm4. In this case the term in R-8 in <a,”z)is 4% of that in and the depolarized rotational scattering should be observable. If all the excess intensity in the observed depolarized spec- trum is attributed to induced rotational scattering then A(4neJ-l would be (4.9 & 2) x cm4 for CH4 and (5.2 & 3) x cm4 for CF4. .T m 0 L 100 200 300 0 / cm’ FIG.1.-The calculated translationally broadened induced rotational spectrum of CHI at 295 K.Points are shown at 20 cm-’ intervals on the Stokes side. The profile beyond 200 cm-’ is described by an exponential function exp(-w/wo) with wo = 130 cm-’. The line labelled DID represents the profile of the dipole-induced-dipole contribution as estimated from the data in ref. (8). A. D. Buckingham Adu. Chem. Phys. 1967,12 107. P. Isnard D. Robert and L. Galatry Mol. Phys. 1976,31,1789. GENERAL DISCUSSION The rotational selection rules are AJ = 0 &1 &2 -+3 and the relative intensity distribution of the rotational spectrum may be calculated. For tetrahedral molecules the tensor A transforms in the same way as the octopole moment.Theoretical spectra have been generated by accounting for translational broadening with a lineshape this is an approximation as the contributions appropriate to the DID intera~tion;~* to the pair polarizability have a different R-dependence. For CH4at 295 K (fig. 1) the shape of the calculated spectrum beyond 200 cm'l is consistent with an exponential profile whose characteristic frequency coois 130 cm-l; the observed value8 is 138 cm-I. For CF4at 295 K (fig. 2) the slope of the high-frequency wing of the calculated spec- I 1 I i 50 100 200 dCl.6' n~. 2.-The calculated translationally broadened induced rotational spectrum of CF4 at 295 K. Points are shown at 5 cm-l intervals on the Stokes side. The profile beyond 40 cm-l is described by an exponential function exp(-coo/oo) with coo = 19.3 cm-'.The line labelled DID represents the profile of the dipole-induced-dipole contribution as estimated from the data in ref (7). trum corresponds to coo = 19.3 cm-l while the experimental value8 is 32.2 cm-l. The poorer agreement with experiment for CF4 is understandable. Methane has a large rotational constant and the lines are widely spaced at high frequencies (AJ = 3); their distribution essentially determines the shape of this part of the spectrum. How-ever in CF4the rotational lines are closely spaced and the shape of the spectrum de- pends critically on the broadening. Dr. P. A. Madden (Cambridge University) said I would like to describe a theoreti- cal calculation of the depolarized Rayleigh scattering from liquid argon at 85.4 IS which may be compared with the experimental lineshape at 84 K reported in the paper under discussion.The calculation is of interest in the present context for the light it casts on the mechanism controversy. The theory proceeds from the expression,l for the depolarized light scattering spectrum (k is the scattering vector and cr3 the P. A. Madden Chem. Phys. Letters 1977,472174. GENERAL DISCUSSION frequency). nk and Fxz(kf)are the Fourier transforms of the number density and the xz component of the tensor giving the interaction induced polarizability $xz(r). In the calculation a DID interaction was assumed where the sums run over all particles and i,dr denotes that integration extends over all space excluding a sphere of radius 42 at the origin.Then k k j(ka) (3) Fx,(k)a dreik.rFxz(r)tc3 v5 in whichj is a first order Bessel function. The dynamical properties enter through the correlation function for the bilinear number density fluctuations. An expression for this correlation function was developed using the Mori approach and the correlation functions appearing in the theory related to parameters obtained from inelastic neu- tron scattering data on liquid argon by use of the Kirkwood Superposition Approxi- mation. The final expression then contains no adjustable parameters and the cal- culated lineshape is in excellent agreement with the experimental one for frequencies 0-100 crn-l. The relevance of this calculation to the mechanism controversy under discussion is that the whole lineshape here arises from a single mechanism and is a property of the dynamics and not the detailed form of the interaction tensor.This conclusion can be seen from the expression for the intensity within the simplified electrodynamics used here The static correlation function appearing is in the Superposition Approximation (S(k’)-1)2 where S(k) is the neutron scattering structure factor. This factor in the integrand is sharply peaked at k‘ III 2n/a and the factor [k’jl(k’a)/(k’a)12 [see eqn (3)] makes the integrand die for all larger k’. All contributions to the spectrum then are determined by the value of the integrand in (1) at k’ N 2n/a. The functional form of Txz(k’), which enters through the first factor on the r.h.s.of (3) matters little since only the value ?xz(27c/a) is important and then only in determining the intensity. Dr. D. W. Oxtoby (Paris)said Depolarized light scattering intensities and spectra depend on the form of the pair polarizability anisotropy P((R). In the lowest order point dipole (DID) approximation p(R) is given by where R is the interatomic distance and a the atomic polarizability. W. M. Gelbart and I have shown’ that when short range effects due to overlap exchange and finite atomic size are taken into account the resulting P(R)can be represented in the form W.M. Gelbart and D. W.Oxtoby Mol. Phys. 1975,29,1569; 1975,30,535. GENERAL DISCUSSION where Ro z a&. Since the exponential term is of shorter range than the point dipole term Litovitz et al.suggest that it will introduce primarily high frequency Fourier components in the light scattering spectrum. In fact exactly the reverse is true because the short range term is subtracted the resulting p(R)varies less steeply with distance and it is low frequency components which are enhanced. This shows that the short and long range parts must be treated together and their influence on spectral features may not be separated. Prof. K. Singer (Royal Holloway College) said We have applied the continued fraction memory function analysis to rotational self correlation functions obtained in molecular dynamics simulations of linear molecules (F2,C12 Br, C02) based on the two centre Lennard-Jones potential .' For these model liquids under conditions corresponding to the experimental triple point the situation is much simpler than in the systems described by Gerschel; that is the higher memory functions are both shorter lived and simpler than the lower ones.To avoid the numerical difficulties which arise in the calculation of higher memory functions from numerical data we have assumed analytical forms with variable para- meters for the highest memory function and optimised the parameters so as to obtain the best fit to the given correlation functions. In addition the following simplifying assumptions are made the third memory function of the orientational autocorrelation (a.c.f.) functions for PI (C,(t))and P2 (C2(t)) and the second memory function of the angular momentum a.c.f.(CJ(t)) differ only by multiplicative factors i.e. M3(2)(t)= MJ2)(0)F(t) for C2(t) (2) M2(J)(t)= MJJ)(0)F(t) for CJ(t). (3) It follows from (3) that the first memory function of the torque a.c.f. ( CT)is M"T'(t) = hf"T'(O)F(t) + M,'J'(O). (4) The zero-time values of the memory functions are determined by the equilibrium prop- erties of the liquid. The trial functions F(t) included (a) exp (-at2) (b) a exp (-alt2) + (1 -a,) exp (-a2?') (c) exp (-at2) + at2 exp (-bt) (d) aexp(-at)(l + at) + (1 -a)exp (-bt)(l + bt). Table 1 lists the standard deviations over the range 0-2 x s between the molecular dynamics results for the four rotational a.c.f.s. and the correlation functions generated from the optimized memory functions.The results for II(b) are shown in graphical form (fig. 1). In each case the initial decay is faster and the size of the positive or negative tail smaller as one goes from a lower to a higher memory func- tion. On the time scale of Cl(t)and C2(t) the first memory function of CTor the second memory function of CJare nearly delta functions. It should be noted however that even these memory functions have a definite small and not very short lived dip (E -0.02 at 0.2 ps and N -0.01 at 0.6 ps) before the asymptotic value is attained. K. Singer A. J. Taylor and J. V. L. Singer unpublished. The computer program and some of the basic ideas in this approach are due to E. Detyna. GENERAL DISCUSSION TABLE 1 standard deviations.system C20) GO) Cdt) T* = 1.06 0.030 0.029 0.023 p* = 0.608 0.008 0.010 0.003 I* = 0.505 0.005 0.013 0.012 T* = 1.00 0.032 0.030 0.031 p* = 0.539 0.004 0.007 0.004 I* = 0.63 0.006 0.009 0.007 III (CO,) T* = 1.345 0.017 0.012 0.009 p* = 0.422 0.001 0.008 0.004 I* = 0.793 0.4-\\ ‘? 0.2-t, \ \‘*-.-.-.-* -. -. -.-.-.-._._._._._._,_._. -__ - - -------O-‘I ,~-‘L// I I I I I 1 I 1 I -0.2 -0.2I I I 1 I I I I 1 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 t I 10-’z’2s FIG.1(b).-Cz(t) the normalised a.c.f. of P2,--the normalised first memory function of C2(t),-* -the normalised second memory function of C2(f) GENERAL D I S CU S S I ON 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 t 1 10-’2s FIG.l(c).-CJ(t) the normalised angular momentum a.c.f.,- -the normalised first memory function of CJ(t),--the normalised second memory function of CJ(t).t I lo-’ FIG.l(d).-cT(t) the normalised torque a.c.f.,- -the normalised first memory function of CT(f). (The discrepancy between the regenerated and given a.c.f.s. would in these graphs only show up as a thickening of the line in some places and has therefore been omitted.) Dr. D. Frenkel (Amsterdam) said In one respect the memory functions for rota- tional motion differ fundamentally from the corresponding functions for translational motions. In the limit of free rotation the former memory functions do not reduce to any simple form whereas in a dilute gas the velocity memory function tends to zero. To give a specific example the dipole correlation function of a linear free rotor is proportional to <u(O) u(t)) = (I,,kT)/mdwco exp (-Iit1~/2kT) 0 GENERAL DISCUSSION the frequency spectrum of the first memory function corresponding to this correlation function is not just complicated but even singular.My question to Gerschel is whether knowing that translation and rotation memory functions behave qualitatively differently at low densities one should attach much significance to similarities at inter- mediate densities ? Dr. G. Wyllie (University of Glasgow) said I should like to express admiration of the experimental work which has enabled Gerschel and his colleagues to obtain such reliable information about the polarization autocorrelation functions.The discus- sion in terms of the well-known Mori hierarchy of equations is illuminating but the equations quoted by Gerschel as (1.1) and (1.2) are not the only ones possible and may be in a certain sense misleading. Ford Kac and Mazurl derived for a limited class of systems an equation of the form 4t) = -r(t)u(t> + n(t) or g(t>= -y(tlg(t) and Tokuyama and Mori recently gave a general derivation. The form is obviously inconvenient if g vanishes anywhere where 8 does not. The autocorrelation function of n has the neat form It is clear from Gerschel’s (1.2) and (2.7) that the immediate physical function g and the autocorrelation function K,of the “ random ” forcefare not linearly related and it can be easily verified for simple linear systems with well defined normal modes that K,typically contains frequencies with which nothing in the system actually vi- brates.The spectra of (f(O)f(t))or of <n(O)n(t)> should be used with caution in the interpretation of molecular mechanisms. Prof. A. Gerschel (Orsay)said In agreement with Frenkel’s remark a close com- parison of the details of translational and rotational memory functions (or 2nd order memory functions) would be irrelevant. Only those important features such as the relatively slow decay and the presence of obvious positive correlations do deserve discussion. If these are physically meaningful in the sense that I developed in my paper then analogies with the findings of translational dynamics deserve to be em- phasized in so far as they are predictions from numerical simulation while here we deal with the first experimental evidence of the effects.Moreover it is interesting to find out similarities in both dynamics translational and rotational since basically the same processes are responsible for elementary displacements either linear or angular. In this respect I should add that a critical examination of model descrip- tions of the orientational motion have led us to prefer the rotational analogue of the “itinerant oscillator ” model a model that also succeeded in describing translational motion as it was primarily intended to In a sense Wyllie’s remark is of a similar nature pointing out the difficulties in assessing the details of memory functions-or their frequency spectra-as definite molecular mechanisms.Since our K, (t) present long lived correlations it may be significant that a Langevin generalized formalism is not well suited to functions like G. W. Ford M. Kac and P. Mazur J. Math. Phys. 1965,6,504. M. Tokuyama and H. Mori Prog. Theor.Phys. 1976,55,411. A. Gerschel I. Dimicoli J. Jaffrd and A. Riou Mul. Phys. 1976 32 679. 178 GENERAL DISCUSSION g, (t)depicting short-time processes the separation into a generalized friction kernel and a random force is no longer meaningful if the fluctuations of the driving force and those of the random force occur in the same time scale. In other words if fluctua- tions in the rotational velocity and fluctuations in the torques responsible for the velocity changes exhibit comparable time dependence the orthogonality condition is no longer satisfied.For these reasons we confined our analysis to sketching striking similarities with molecular dynamics results refraining from inferring much more detailed motional mechanisms from our second memory function characteristics. Kivelson remarked that a mathematical proof of the identification of functions Kg(t) and grY(t)has been already provided with further identification of both these func- tions with the angular velocity correlation function. Such proofs have been given by many authors including ourselves provided that the orientation correlation functions decay slowly enough in comparison with the angular velocity (or the rotational velocity) correlation functi0n.l It was to test how severe that restriction was that we system- atically computed both functions Kg(t)and g,,(t) under varied physical states of a same liquid resulting in quite satisfactory agreement at high densities and a gradual de- parture for lower densities states.Prof. A. D. Buckingham (Cambridge Uniuersity) asked What is the angle-depend- ent pair potential employed by van der Elsken and Frenkel in their calculations of the rotational line-widths for HCI in compressed Ar and how does it compare with potentials deduced from information about the HClAr "molecule "? Dr. D. Frenkel (Amsterdam) said:In the molecular dynamics calculations described in the present paper the anisotropic perturbation acting on a probe molecule in a dense fluid was computed. In these calculations it was assumed that the r-dependence of the I = 1 and I = 2 part of the anisotropic probe-host interaction was proportional to the Ar-Ar Lennard-Jones (6-12) potential (a = 3.405 A).Clearly this will be an oversimplified description of the HC1-argon anisotropic interaction. For instance the long-range behaviour of the HCI-argon interaction that should go as r-' is described by Y-~. Moreover one may expect the position of the minimum of the I = 1,2 parts of the potential to differ somewhat from rmi 2% for the L.J. (6-12) potential. However it should be realized that the quite detailed investigations of the HC1-argon anisotropic interaction by Neilsen and Gordon,2 Dunker and Gordon3 and Holmgren and Klemperer4 have not yet provided us with a set of anisotropic potential parameters that account for all experimental information on the binary HCl-Ar interaction.In fact the different potentials that have been proposed differ in their estimate of the depth of the I = 1 (I = 2) part by a factor of 2 to 3. In particular it is not known whether the I = 1 or the I = 2 interaction has the deeper minimum. In view of these observations there is little reason to reject the anisotropic potentials that are used in the present M.D. calculations as being inadequate to describe known characteristics of the HCI-Ar anisotropic interaction. It should be mentioned that the values for the depth of the I = 1 and I = 2 part of the anisotropic interaction that yielded the best fit for the linewidths given in table 1; cf.cl = 32.3 cm-l and e2 = 49.1 cm-' fall within the range of values given in ref. (1) and (3). However our estimates for cl and c2 should be treated cautiously because they were obtained by A. Gerschel I. Darmon and C. Brot MoZ. Phys. 1972,23 317. W. B. Neilsen and R. G. Gordon J. Chem. Phys. 1973,58,4131. A. M. Dunker and R. G. Gordon J. Chem. Phys. 1976,64,354. S. L. Holmgren and W. Klemperer personal communication. GENERAL DISCUSSION fitting experimental data on dense gases. Non-pairwise additive contributions to the anisotropic interaction may be of some importance. Hence we may only hope to obtain information about an effective two-body anisotropic interaction. Prof. A. D. Buckingham (Cambridge University) made the following summarizing remarks We have focused our attention in this Symposium on some of the newer techniques for studying the relaxation of molecules.In particular we have heard papers on dielectric polarization and the Kerr effect on polarized and depolarized light scattering on resonance Raman scattering on infrared absorption and on com- puter simulation by molecular dynamics calculations. Other techniques such as inelastic neutron scattering and magnetic resonance spectroscopy would also have been appropriate but were excluded by the organizing Committee who sought to arrange a compact and coherent Symposium. Non-linear polarization and scattering were not excluded but have not featured in our discussion; a paper in this area was not presented owing to the regrettable absence of Professor Kielich.I shall not attempt to summarize the individual papers-the authors have done that for themselves-but shall make some general remarks. The first is to emphasize a point made in one or two of the manuscripts-we should strive to employ several techniques on the same system. Each of the techniques we have heard about has its special virtues but none in isolation provides a full story. The Symposium has been helpful in the way it has brought together several complementary techniques; let us hope that it may lead to joint investigations of the same system in different laboratories. And we should strive to exploit as many variables as possible; density changes are particularly to be recommended and fortunately high-pressure instrumentation is now simple and convenient.It may be useful to divide our subject into three parts the molecules the inter- molecular forces and the bulk phase and for each of us to ask ourselves the awkward question as to where our principal interest lies. If we are chiefly concerned about the structure and properties of isolated molecules then we should direct our efforts to the gaseous or crystalline phases and avoid non-equilibrium phenomena. We may be interested in changes in molecular properties in going from the isolated to the dissolved state; that would compel us to look at liquids and solutions but we should concen- trate on those techniques such as vibrational spectroscopy where the excitation is highly localized and bear in mind that the identity of a molecule may be lost in a condensed medium.Where does one molecule end and another begin? If our main interest is in intermolecular forces we should again try to work with gases at low densities or with crystals and exploit the advantages of different techniques. Thus molecular-beam scattering combined with second-virial coefficients or with calculated long-range interaction energies can give us accurate potentials for simple systems such as the inert gases and HZ.And double-resonance spectroscopy in gases can yield the probability that a collision causes a transition between particular quantum states; tuneable lasers will extend the range of this particular technique. On the other hand if our interests are in the fluid phase itself we shall wish to exploit a number of tech- niques and become involved with models of reality.Information obtained in one area can be fruitful in another-thus knowledge of the properties of isolated molecules provides us with a description of the long-range intermolecular forces and a know- ledge of these and of the short-range forces (where electron exchange renders the integrity of the individual molecules doubtful) illuminates the mechanisms of energy transfer and molecular relaxation. The technique of computer simulation provides us with the opportunity to unravel the complex nature of molecular interactions and relaxation in dense fluids. It should GENERAL DISCUSSION help us to gain a firmer understanding of the relative significance of different micro-scopic contributions to an observable.The technique has already answered some questions and posed some new ones and is bound to grow in importance. As it will soon be Christmas it seems appropriate to end by setting a puzzle con- cerning the rotational motion of a polar body in a fluid. We know that a dipole p in a spherical cavity in a continuum induces in the cavity a reaction field R proportional to p R =gp. In Onsager’s theory of dielectricsi g = 2(~-1)[(2e + 1)47~q,a~]-~ where E~ is the permittivity of free space E is the dielectric constant of the continuum and a is the radius of the cavity. With a dipole inside an ellipsoid of revolution the principal components of the reaction field are R1 = gipi R2 =gzlU2 where g # g2. In general therefore the reaction field is not parallel to p and the dipole experiences a torque p x R of magnitude plp2(g -gl).If the dipole is fixed in the ellipsoid it might be supposed that the whole will rotate and give rise to per- petual motion! Those attending the Symposium were left to solve the paradox. The solution is to be found by seeking an equal and opposite torque on the ellipsoid; its source lies in the asymmetry in the energy density of the continuum in which there is an electric field due to p and to the boundary of the ellipsoid. L. Onsager J. Amer. Chem. Soc. 1936,58 1486.

ISSN:0301-5696    

DOI:10.1039/FS9771100148

出版商:RSC    

年代:1977

数据来源: RSC